Package 'matchingMarkets' reference manual

Title:	Analysis of Stable Matchings
Description:	Implements structural estimators to correct for the sample selection bias from observed outcomes in matching markets. This includes one-sided matching of agents into groups as well as two-sided matching of students to schools. The package also contains algorithms to find stable matchings in the three most common matching problems: the stable roommates problem, the college admissions problem, and the house allocation problem.
Authors:	Thilo Klein [aut, cre], Robert Aue [ctb], Sven Giegerich [ctb], Alexander Sauer [ctb]
Maintainer:	Thilo Klein <[email protected]>
License:	GPL (>= 2)
Version:	1.0-2
Built:	2025-03-06 04:35:34 UTC
Source:	https://github.com/thiloklein/matchingmarkets

An R package for the analysis of stable matchings.

Description

The matchingMarkets package contains R, C++ and Java code for stable matching algorithms and the estimation of structural models that correct for the sample selection bias of observed outcomes in matching markets.

Matching is concerned with who transacts with whom, and how. For example, who works at which job, which students go to which school, who forms a workgroup with whom, and so on.

The empirical analysis of matching markets is naturally subject to sample selection problems. If agents match assortatively on characteristics unobserved to the analyst but correlated with both the exogenous variable and the outcome of interest, regression estimates will generally be biased.

The matchingMarkets package comprises

Bayes estimators. The estimators implemented in function stabit and stabit2 correct for the selection bias from endogenous matching.

The current package version provides solutions for two commonly observed matching processes: (i) the group formation problem with fixed group sizes and (ii) the college admissions problem. These processes determine which matches are observed – and which are not – and this is a sample selection problem.
Post-estimation tools. Setting mfx=TRUE in the summary function computes marginal effects from coefficients in binary outcome and selection equations and khb implements the Karlson-Holm-Breen test for confounding due to sample selection.
Design matrix generation. The estimators are based on independent variables for all feasible, i.e., observed and counterfactual, matches in the market. Generating the characteristics of all feasible matches from individual-level data is a combinatorial problem. The package returns design matrices based on pre-specified transformations to generate counterfactual matches.
Algorithms. The package also contains matching algorithms that can be used to simulated matching data: hri: A constraint model (Posser, 2014) for the stable marriage and college admissions problem, a.k.a. hospital/residents problem (see Gale and Shapley, 1962). sri: A constraint model for the stable roommates problem (see Gusfield and Irving, 1989). ttc: The top-trading-cycles algorithm for the housing market problem. These can be used to obtain stable matchings from simulated or real preference data (see Shapley and Scarf, 1974).
Data. In addition to the baac00 dataset from borrowing groups in Thailands largest agricultural lending program, the package provides functions stabsim and stabsim2 to simulate one's own data from one-sided and two-sided matching markets.

Frequently Asked Questions

Why can I not use the classic Heckman correction?

Estimators such as the Heckman (1979) correction (in package sampleSelection) or double selection models are inappropriate for this class of selection problems. To see this, note that a simple first stage discrete choice model assumes that an observed match reveals match partners' preferences over each other. In a matching market, however, agents can only choose from the set of partners who would be willing to form a match with them and we do not observe the players' relevant choice sets.
Do I need an instrumental variable to estimate the model?

Short answer: No. Long answer: The characteristics of other agents in the market serve as the source of exogenous variation necessary to identify the model. The identifying exclusion restriction is that characteristics of all agents in the market affect the matching, i.e., who matches with whom, but it is only the characteristics of the match partners that affect the outcome of a particular match once it is formed. No additional instruments are required for identification (Sorensen, 2007).
What are the main assumptions underlying the estimator?

The approach has certain limitations rooted in its restrictive economic assumptions.
1. The matching models are complete information models. That is, agents are assumed to have a complete knowledge of the qualities of other market participants.
2. The models are static equilibrium models. This implies that (i) the observed matching must be an equilibrium, i.e., no two agents would prefer to leave their current partners in order to form a new match (definition of pairwise stability), and (ii) the equilibrium must be unique for the likelihood function of the model to be well defined (Bresnahan and Reiss, 1991).
3. Uniqueness results can be obtained in two ways. First, as is common in the industrial organization literature, by imposing suitable preference restrictions. A suitable restriction on agents' preferences that guarantees a unique equilibrium is alignment (Pycia, 2012). In a group formation model, (pairwise) preference alignment states that any two agents who belong to the same groups must prefer the same group over the other. Second, by choosing a market assigment based on matching algorithms that produce a unique stable matching, such as the well-studied Gale and Shapley (1962) deferred acceptance algorithm.
4. Finally, the models assume bivariate normality of the errors in selection and outcome equation. If that assumption fails, the estimator is generally inconsistent and can provide misleading inference in small samples.

How to cite this package

Whenever using this package, please cite as

Klein, T. (2018). matchingMarkets: Structural Estimator and Algorithms for the Analysis of Stable Matchings. R package version 1.0-1.

Author(s)

Thilo Klein

References

Bresnahan, T. and Reiss, P. (1991). Empirical models of discrete games. Journal of Econometrics, 48(1-2):57–81.

Gale, D. and Shapley, L.S. (1962). College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9–15.

Gusfield, D.M. and R.W. Irving (1989). The stable marriage problem: Structure and algorithms, MIT Press.

Heckman, J. (1979). Sample selection bias as a specification error. Econometrica, 47(1):153–161.

Prosser, P. (2014). Stable Roommates and Constraint Programming. Lecture Notes in Computer Science, CPAIOR 2014 Edition. Springer International Publishing, 8451: 15–28.

Pycia, M. (2012). Stability and preference alignment in matching and coalition formation. Econometrica, 80(1):323–362.

Shapley, L. and H. Scarf (1974). On cores and indivisibility. Journal of Mathematical Economics, 1(1):23–37.

Sorensen, M. (2007). How smart is smart money? A two-sided matching model of venture capital. The Journal of Finance, 62(6):2725–2762.

Townsend Thai Project BAAC Annual Resurvey, 2000

Description

The baac00 data frame contains data of 292 borrowers from Thailand's largest agricultural lending program. These data are collected as part of the Townsend Thai Project Bank for Agriculture and Agricultural Cooperatives (BAAC) Annual Resurvey (Townsend, 2000). The 292 borrowers are nested within 68 groups and 39 markets. This nestedness makes the dataset particularly relevant for matching applications. A more complete discussion of the data is found in Ahlin (2009), Section 3, and Klein (2015a).

Usage

data(baac00)data(baac00)

Format

This data frame contains the following columns:

g.id: group identifier.
m.id: market identifier.
R: repayment outcome: BAAC never raised interest rate as a penalty for late repayment.
pi: success probability: measure of group members' project success probability.
wst: worst year: indicator of economically worst year. 1:last year; 2:year before last year; 101-168:neither.
loan_size: loan size: average loan size borrowed by the group.
loan_size2: loan size squared.
lngroup_agei: log group age: log of number of years group has existed.

Source

Townsend, R. (2000). Townsend Thai Project Bank for Agriculture and Agricultural Cooperatives (BAAC) Annual Resurvey, 2000. Available at http://hdl.handle.net/1902.1/12057, Murray Research Archive.

References

Ahlin, C. (2009). Matching for credit: Risk and diversification in Thai microcredit groups. Working Paper 251, Bureau for Research and Economic Analysis of Development.

Klein, T. (2015a). Does Anti-Diversification Pay? A One-Sided Matching Model of Microcredit. Cambridge Working Papers in Economics, #1521.

All stable matchings in the hospital/residents problem with incomplete lists

Description

Finds all stable matchings in either the hospital/residents problem (a.k.a. college admissions problem) or the related stable marriage problem. Dependent on the problem, the results comprise the student and college-optimal or the men and women-optimal matchings. The implementation allows for incomplete preference lists (some agents find certain agents unacceptable) and unbalanced instances (unequal number of agents on both sides). The function uses the Prosser (2014) constraint encoding based on either given or randomly generated preferences.

Usage

hri(
  nStudents = ncol(s.prefs),
  nColleges = ncol(c.prefs),
  nSlots = rep(1, nColleges),
  s.prefs = NULL,
  c.prefs = NULL,
  s.range = NULL,
  c.range = NULL,
  randomization = NULL,
  seed = NULL,
  check_consistency = TRUE,
  ...
)
hri(
  nStudents = ncol(s.prefs),
  nColleges = ncol(c.prefs),
  nSlots = rep(1, nColleges),
  s.prefs = NULL,
  c.prefs = NULL,
  s.range = NULL,
  c.range = NULL,
  randomization = NULL,
  seed = NULL,
  check_consistency = TRUE,
  ...
)

Arguments

`nStudents`	integer indicating the number of students (in the college admissions problem) or men (in the stable marriage problem) in the market. Defaults to `ncol(s.prefs)`.
`nColleges`	integer indicating the number of colleges (in the college admissions problem) or women (in the stable marriage problem) in the market. Defaults to `ncol(c.prefs)`.
`nSlots`	vector of length `nColleges` indicating the number of places (i.e. quota) of each college. Defaults to `rep(1,nColleges)` for the marriage problem.
`s.prefs`	matrix of dimension `nColleges` `x` `nStudents` with the `j`th column containing student `j`'s ranking over colleges in decreasing order of preference (i.e. most preferred first).
`c.prefs`	matrix of dimension `nStudents` `x` `nColleges` with the `i`th column containing college `i`'s ranking over students in decreasing order of preference (i.e. most preferred first).
`s.range`	range of two intergers `s.range = c(s.min, s.max)`, where `s.min < s.max`. Produces incomplete preference lists with the length of each student's list randomly sampled from the range `[s.min, s.max]`. Note: interval is only correct if either c.range or s.range is used.
`c.range`	range of two intergers `c.range = c(c.min, c.max)`, where `c.min < c.max`. Produces incomplete preference lists with the length of each college's list randomly sampled from the range `[c.min, c.max]`. Note: interval is only correct if either c.range or s.range is used.
`randomization`	determines at which level random lottery numbers for student priorities are drawn. The default is `randomization = "multiple"`, where a student's priority is determined by a separate lottery at each college (i.e. local tie-breaking). For the second variant, `randomization = "single"`, a single lottery number determines a student's priority at all colleges (i.e. global tie breaking).
`seed`	integer setting the state for random number generation.
`check_consistency`	Performs consicentcy checks (Checks if there are columns in the preference matrices that only contains zeros and drops them and checks the matrixes for consistencies if they are given by characters). Defaults to `TRUE` but changing it to `FALSE` might reduce the running-time for large problems.
`...`	.

Value

hri returns a list of the following elements.

`s.prefs.smi`	student-side preference matrix for the stable marriage problem with incomplete lists (SMI).
`c.prefs.smi`	college-side preference matrix for the stable marriage problem with incomplete lists (SMI).
`s.prefs.hri`	student-side preference matrix for the college admissions problem (a.k.a. hospital/residents problem) with incomplete lists (HRI).
`c.prefs.hri`	college-side preference matrix for the college admissions problem (a.k.a. hospital/residents problem) with incomplete lists (HRI).
`matchings`	edgelist of matched students and colleges, inculding the number of the match (`matching`) and two variables that indicate the student-optimal match (`sOptimal`) and college-optimal match (`cOptimal`)

Minimum required arguments

hri requires the following combination of arguments, subject to the matching problem.

nStudents, nColleges: Marriage problem with random preferences.
s.prefs, c.prefs: Marriage problem with given preferences.
nStudents, nSlots: College admissions problem with random preferences.
s.prefs, c.prefs, nSlots: College admissions problem with given preferences.

Author(s)

Thilo Klein

References

Gale, D. and L.S. Shapley (1962). College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9–15.

Morizumi, Y., T. Hayashi and Y. Ishida (2011). A network visualization of stable matching in the stable marriage problem. Artificial Life Robotics, 16:40–43.

Prosser, P. (2014). Stable Roommates and Constraint Programming. Lecture Notes in Computer Science, CPAIOR 2014 Edition. Springer International Publishing, 8451: 15–28.

Examples

## Not run: 
## -----------------------
## --- Marriage problem 

## 7 men, 6 women, random preferences:
 hri(nStudents=7, nColleges=6, seed=4)

## 3 men, 2 women, given preferences:
 s.prefs <- matrix(c(1,2, 1,2, 1,2), 2,3)
 c.prefs <- matrix(c(1,2,3, 1,2,3), 3,2)
 hri(s.prefs=s.prefs, c.prefs=c.prefs)

## 3 men, 2 women, given preferences:
 s.prefs <- matrix(c("x","y", "x","y", "x","y"), 2,3)
 colnames(s.prefs) <- c("A","B","C")
 c.prefs <- matrix(c("A","B","C", "A","B","C"), 3,2)
 colnames(c.prefs) <- c("x","y")
 hri(s.prefs=s.prefs, c.prefs=c.prefs)

## --------------------------------
## --- College admission problem 

## 7 students, 2 colleges with 3 slots each, random preferences:
 hri(nStudents=7, nSlots=c(3,3), seed=21)

## 7 students, 2 colleges with 3 slots each, given preferences:
 s.prefs <- matrix(c(1,2, 1,2, 1,NA, 1,2, 1,2, 1,2, 1,2), 2,7)
 c.prefs <- matrix(c(1,2,3,4,5,6,7, 1,2,3,4,5,NA,NA), 7,2)
 hri(s.prefs=s.prefs, c.prefs=c.prefs, nSlots=c(3,3))
 
## 7 students, 2 colleges with 3 slots each, given preferences:
 s.prefs <- matrix(c("x","y", "x","y", "x",NA, "x","y", 
                     "x","y", "x","y", "x","y"), 2,7)
 colnames(s.prefs) <- c("A","B","C","D","E","F","G")
 c.prefs <- matrix(c("A","B","C","D","E","F","G", 
                     "A","B","C","D","E",NA,NA), 7,2)
 colnames(c.prefs) <- c("x","y")
 hri(s.prefs=s.prefs, c.prefs=c.prefs, nSlots=c(3,3))
 
## 7 students, 3 colleges with 3 slots each, incomplete preferences:
 hri(nStudents=7, nSlots=c(3,3,3), seed=21, s.range=c(1,3))
 
 s.prefs <- matrix(c('S1', 'S2', NA,
                     'S3', 'S1', NA,
                     'S1', NA, NA,
                      NA, NA,NA,
                     'S2', 'S1', 'S5'),
                   nrow = 3, ncol = 5)
 # Note that we explicitly allow for the existence of entries refering to colleges
 # that do not exist. A warning is generated and the entry is ignored.
 colnames(s.prefs) <- c('A', 'B', 'C', 'D', 'E')
 c.prefs <- matrix(c('B', 'C','D', 'A',
                     'C', 'D', NA, NA, 
                     'D', 'B', 'A', 'E'),
                   nrow = 4, ncol = 3)
 colnames(c.prefs) <- c('S1', 'S2', 'S3')
 hri(s.prefs=s.prefs, c.prefs=c.prefs, nSlots=c(3,3,3), check_consistency = TRUE)
## --------------------
## --- Summary plots

## 200 students, 200 colleges with 1 slot each
 res <- hri(nStudents=200, nColleges=200, seed=12)
 plot(res)
 plot(res, energy=TRUE)

## End(Not run)
## Not run: 
## -----------------------
## --- Marriage problem 

## 7 men, 6 women, random preferences:
 hri(nStudents=7, nColleges=6, seed=4)

## 3 men, 2 women, given preferences:
 s.prefs <- matrix(c(1,2, 1,2, 1,2), 2,3)
 c.prefs <- matrix(c(1,2,3, 1,2,3), 3,2)
 hri(s.prefs=s.prefs, c.prefs=c.prefs)

## 3 men, 2 women, given preferences:
 s.prefs <- matrix(c("x","y", "x","y", "x","y"), 2,3)
 colnames(s.prefs) <- c("A","B","C")
 c.prefs <- matrix(c("A","B","C", "A","B","C"), 3,2)
 colnames(c.prefs) <- c("x","y")
 hri(s.prefs=s.prefs, c.prefs=c.prefs)

## --------------------------------
## --- College admission problem 

## 7 students, 2 colleges with 3 slots each, random preferences:
 hri(nStudents=7, nSlots=c(3,3), seed=21)

## 7 students, 2 colleges with 3 slots each, given preferences:
 s.prefs <- matrix(c(1,2, 1,2, 1,NA, 1,2, 1,2, 1,2, 1,2), 2,7)
 c.prefs <- matrix(c(1,2,3,4,5,6,7, 1,2,3,4,5,NA,NA), 7,2)
 hri(s.prefs=s.prefs, c.prefs=c.prefs, nSlots=c(3,3))
 
## 7 students, 2 colleges with 3 slots each, given preferences:
 s.prefs <- matrix(c("x","y", "x","y", "x",NA, "x","y", 
                     "x","y", "x","y", "x","y"), 2,7)
 colnames(s.prefs) <- c("A","B","C","D","E","F","G")
 c.prefs <- matrix(c("A","B","C","D","E","F","G", 
                     "A","B","C","D","E",NA,NA), 7,2)
 colnames(c.prefs) <- c("x","y")
 hri(s.prefs=s.prefs, c.prefs=c.prefs, nSlots=c(3,3))
 
## 7 students, 3 colleges with 3 slots each, incomplete preferences:
 hri(nStudents=7, nSlots=c(3,3,3), seed=21, s.range=c(1,3))
 
 s.prefs <- matrix(c('S1', 'S2', NA,
                     'S3', 'S1', NA,
                     'S1', NA, NA,
                      NA, NA,NA,
                     'S2', 'S1', 'S5'),
                   nrow = 3, ncol = 5)
 # Note that we explicitly allow for the existence of entries refering to colleges
 # that do not exist. A warning is generated and the entry is ignored.
 colnames(s.prefs) <- c('A', 'B', 'C', 'D', 'E')
 c.prefs <- matrix(c('B', 'C','D', 'A',
                     'C', 'D', NA, NA, 
                     'D', 'B', 'A', 'E'),
                   nrow = 4, ncol = 3)
 colnames(c.prefs) <- c('S1', 'S2', 'S3')
 hri(s.prefs=s.prefs, c.prefs=c.prefs, nSlots=c(3,3,3), check_consistency = TRUE)
## --------------------
## --- Summary plots

## 200 students, 200 colleges with 1 slot each
 res <- hri(nStudents=200, nColleges=200, seed=12)
 plot(res)
 plot(res, energy=TRUE)

## End(Not run)

Resident-optimal matching in the hospital/residents problem with couples

Description

Implements the Roth Peranson matching algorithm for the hospital/residents problem with couples as described in Roth and Peranson (1999). The function is based on an adoption of Bacchus (2018).

Usage

hri2(
  nStudents = ncol(s.prefs),
  nColleges = ncol(c.prefs),
  nSlots = rep(1, nColleges),
  nCouples = ncol(co.prefs),
  s.prefs = NULL,
  c.prefs = NULL,
  co.prefs = NULL,
  randomization = "multiple",
  seed = NULL,
  check_consistency = TRUE,
  ...
)
hri2(
  nStudents = ncol(s.prefs),
  nColleges = ncol(c.prefs),
  nSlots = rep(1, nColleges),
  nCouples = ncol(co.prefs),
  s.prefs = NULL,
  c.prefs = NULL,
  co.prefs = NULL,
  randomization = "multiple",
  seed = NULL,
  check_consistency = TRUE,
  ...
)

Arguments

`nStudents`	integer indicating the number of students (in the college admissions problem) or men (in the stable marriage problem) in the market. Defaults to `ncol(s.prefs)`.
`nColleges`	integer indicating the number of colleges (in the college admissions problem) or women (in the stable marriage problem) in the market. Defaults to `ncol(c.prefs)`.
`nSlots`	vector of length `nColleges` indicating the number of places (i.e. quota) of each college. Defaults to `rep(1,nColleges)` for the marriage problem.
`nCouples`	integer indicating the number of couples (in the college admissions problem) or men (in the stable marriage problem) in the market. Defaults to `ncol(co.prefs)`
`s.prefs`	matrix of dimension `nColleges` `x` `nStudents` with the `j`th column containing student `j`'s ranking over colleges in decreasing order of preference (i.e. most preferred first).
`c.prefs`	matrix of dimension `nStudents` `x` `nColleges` with the `i`th column containing college `i`'s ranking over students in decreasing order of preference (i.e. most preferred first).
`co.prefs`	matrix of dimension `4` `x` `nCouplesPrefs` in long format with the `1`th and `2`th columns containing student couple id's; `3`th and `4`th is a 2-tuple ranking over college preference for the couple (coupleStudent1.pref, coupleStudent2.pref) in decreasing order of preference by rows (i.e. most preferred first).
`randomization`	determines at which level and in which order random lottery numbers for student priorities are drawn. The default is `randomization = "multiple"`, where a student's priority is determined by a separate lottery at each college (i.e. local tie-breaking). For the second variant, `randomization = "single"`, a single lottery number determines a student's priority at all colleges (i.e. global tie breaking). A third variant is common in the context of course allocation, where a "couple" represents a student who submits a preference ranking over single courses (first course) and combinations of courses (first and second course). Here, the option `randomization = "single-course-first"` gives applications for a student's single courses strictly higher priority than for course combinations. This ensures the fairness criterion that a student is only assigned a second course after single course applications of all students have been considered.
`seed`	integer setting the state for random number generation.
`check_consistency`	Performs additional consicentcy checks if the preference matrices are given by characters. Defaults to `FALSE`. Set to `FALSE` to reduce run-time.
`...`	.

Value

hri2 returns a list of the following elements:

`matchings`	List of matched students and colleges.
`summary`	Detailed report of the matching result, including futher information on ranks.

Minimum required arguments

hri2 requires the following combination of arguments, subject to the matching problem.

nStudents, nColleges: Residence hospital problem without couples and random preferences
nStudents, nColleges, nCouples, nSlots: Residence hospital problem with couples and random preferences.
s.prefs, c.prefs, co.prefs, nSlots: Residence hospital problem with couples and given preferences.

Author(s)

Sven Giegerich, Thilo Klein

References

Bacchus, F. (2018). Stable matching suite. GitHub repository.

Gale, D. and L.S. Shapley (1962). College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9–15.

Roth, A. E., & Peranson, E. (1999). The redesign of the matching market for American physicians: Some engineering aspects of economic design. American economic review, 89(4), 748-780.

Kojima, F., Pathak, P. A., & Roth, A. E. (2013). Matching with couples: Stability and incentives in large markets. The Quarterly Journal of Economics, 128(4), 1585-1632.

Examples

## Not run: 
## Example with given preferences
(s.prefs <- matrix(c(4,2,3,5, 2,1,3,NA, 1,2,3,4), 4,3))
(c.prefs <- matrix(rep(1:5,5), 5,5))
(co.prefs <- matrix(c(rep(4,3), rep(5,3), 3,3,NA, 3,NA,3), 3,4))
res <- hri2(s.prefs=s.prefs, c.prefs=c.prefs, co.prefs=co.prefs, nSlots=rep(1,5))
res$matchings
# summary(res)

## Example with random preferences
nStudents <- 50
nColleges <- 30
nCouples <- 4
nSlots <- sample(1:nStudents, nColleges)
res <- hri2(nStudents=nStudents, nColleges=nColleges, nCouples=nCouples, nSlots=nSlots)
res$matchings
# summary(res)

## Example with characters in the preferences matrices
s.prefs <- matrix(c("Micro1", NA, NA,
                    "Micro2", "Micro1", "Macro",
                    "Macro",NA ,NA), 
                    ncol = 3)
colnames(s.prefs) <- c('Lea', 'Mia', 'Kai')
c.prefs <- matrix(c("Niklas", "Kai", "Mia", "Anna",
                    "Lea", "Kai", "Anna",NA,
                    "Kai", "Mia", "Lea",NA), 
                    ncol = 3)
colnames(c.prefs) <- c('Micro1', 'Micro2', 'Macro')
col1 <- c(rep("Niklas",4),rep("Anna",5))
col2 <- c(rep("Jan",4),rep("Lisa",5))
col3 <- c("Micro1","Macro","Micro1",NA,"Macro",
          NA,"Micro2","Micro2","Macro")
col4 <- c("Micro2","Micro1",NA,"Macro","Macro",
          "Micro1","Micro2","Macro",NA)
co.prefs <- matrix(c(col1,col2,col3,col4), ncol = 4)
res <- hri2(s.prefs=s.prefs, c.prefs=c.prefs, co.prefs=co.prefs, 
            nSlots=c(2,1,1))                     
res$matching

## Example if students are allowed to apply and be accepted by two courses   
col12 <- c(rep(c(rep("Niklas",4),rep("Anna",2)),2))
col3 <- c("Micro1","Macro","Micro1","Macro","Macro","Macro")
col4 <- c("Micro2","Micro1",NA,NA,"Micro1","Micro2")
co.prefs <- matrix(c(col12,col3,col4), ncol = 4)
res <- hri2(s.prefs=s.prefs, c.prefs=c.prefs, co.prefs=co.prefs, 
            nSlots=c(2,1,1))                     
res$matching

## End(Not run)                                                                                
## Not run: 
## Example with given preferences
(s.prefs <- matrix(c(4,2,3,5, 2,1,3,NA, 1,2,3,4), 4,3))
(c.prefs <- matrix(rep(1:5,5), 5,5))
(co.prefs <- matrix(c(rep(4,3), rep(5,3), 3,3,NA, 3,NA,3), 3,4))
res <- hri2(s.prefs=s.prefs, c.prefs=c.prefs, co.prefs=co.prefs, nSlots=rep(1,5))
res$matchings
# summary(res)

## Example with random preferences
nStudents <- 50
nColleges <- 30
nCouples <- 4
nSlots <- sample(1:nStudents, nColleges)
res <- hri2(nStudents=nStudents, nColleges=nColleges, nCouples=nCouples, nSlots=nSlots)
res$matchings
# summary(res)

## Example with characters in the preferences matrices
s.prefs <- matrix(c("Micro1", NA, NA,
                    "Micro2", "Micro1", "Macro",
                    "Macro",NA ,NA), 
                    ncol = 3)
colnames(s.prefs) <- c('Lea', 'Mia', 'Kai')
c.prefs <- matrix(c("Niklas", "Kai", "Mia", "Anna",
                    "Lea", "Kai", "Anna",NA,
                    "Kai", "Mia", "Lea",NA), 
                    ncol = 3)
colnames(c.prefs) <- c('Micro1', 'Micro2', 'Macro')
col1 <- c(rep("Niklas",4),rep("Anna",5))
col2 <- c(rep("Jan",4),rep("Lisa",5))
col3 <- c("Micro1","Macro","Micro1",NA,"Macro",
          NA,"Micro2","Micro2","Macro")
col4 <- c("Micro2","Micro1",NA,"Macro","Macro",
          "Micro1","Micro2","Macro",NA)
co.prefs <- matrix(c(col1,col2,col3,col4), ncol = 4)
res <- hri2(s.prefs=s.prefs, c.prefs=c.prefs, co.prefs=co.prefs, 
            nSlots=c(2,1,1))                     
res$matching

## Example if students are allowed to apply and be accepted by two courses   
col12 <- c(rep(c(rep("Niklas",4),rep("Anna",2)),2))
col3 <- c("Micro1","Macro","Micro1","Macro","Macro","Macro")
col4 <- c("Micro2","Micro1",NA,NA,"Micro1","Micro2")
co.prefs <- matrix(c(col12,col3,col4), ncol = 4)
res <- hri2(s.prefs=s.prefs, c.prefs=c.prefs, co.prefs=co.prefs, 
            nSlots=c(2,1,1))                     
res$matching

## End(Not run)

Immediate Acceptance Algorithm (a.k.a. Boston mechanism) for two-sided matching markets

Description

Finds the optimal assignment of students to colleges in the college admissions problem based on the Boston mechanism. The algorithmen is also applicable to the stable marriage problem. The option acceptance="deferred" instead uses the Gale-Shapley (1962) Deferred Acceptance Algorithm with student offer. The function works with either given or randomly generated preferences.

Usage

iaa(
  nStudents = ncol(s.prefs),
  nColleges = ncol(c.prefs),
  nSlots = rep(1, nColleges),
  s.prefs = NULL,
  c.prefs = NULL,
  acceptance = "immediate",
  short_match = TRUE,
  seed = NULL
)
iaa(
  nStudents = ncol(s.prefs),
  nColleges = ncol(c.prefs),
  nSlots = rep(1, nColleges),
  s.prefs = NULL,
  c.prefs = NULL,
  acceptance = "immediate",
  short_match = TRUE,
  seed = NULL
)

Arguments

`nStudents`	integer indicating the number of students (in the college admissions problem) or men (in the stable marriage problem) in the market. Defaults to `ncol(s.prefs)`.
`nColleges`	integer indicating the number of colleges (in the college admissions problem) or women (in the stable marriage problem) in the market. Defaults to `ncol(c.prefs)`.
`nSlots`	vector of length `nColleges` indicating the number of places (i.e. quota) of each college. Defaults to `rep(1,nColleges)` for the marriage problem.
`s.prefs`	matrix of dimension `nColleges` `x` `nStudents` with the `j`th column containing student `j`'s ranking over colleges in decreasing order of preference (i.e. most preferred first).
`c.prefs`	matrix of dimension `nStudents` `x` `nColleges` with the `i`th column containing college `i`'s ranking over students in decreasing order of preference (i.e. most preferred first).
`acceptance`	if `acceptance="deferred"` returns the solution found by the student-proposing Gale-Shapley deferred acceptance algorithm; if `acceptance="immediate"` (the default) returns the solution found by the Boston mechanism.
`short_match`	(Optional) If `FALSE` then in the returned matching, free capacities will be indicated with 0 entries. If `TRUE`, free capacities will not be reported in the returned matching but an additonal data.frame is returned that contains free capacities. Defaults to `TRUE`.
`seed`	(Optional) integer setting the state for random number generation.

Value

iaa returns a list with the following elements.

`s.prefs`	student-side preference matrix.
`c.prefs`	college-side preference matrix.
`iterations`	number of interations required to find the stable matching.
`matchings`	edgelist of matches
`singles`	identifier of single (or unmatched) students/men.

Minimum required arguments

iaa requires the following combination of arguments, subject to the matching problem.

nStudents, nColleges: Marriage problem with random preferences.
s.prefs, c.prefs: Marriage problem with given preferences.
nStudents, nSlots: College admissions problem with random preferences.
s.prefs, c.prefs, nSlots: College admissions problem with given preferences.

Author(s)

Thilo Klein

References

Gale, D. and Shapley, L.S. (1962). College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9–15.

Kojima, F. and M.U. Unver (2014). The "Boston" school-choice mechanism. Economic Theory, 55(3): 515–544.

Examples

##\dontrun{
## --------------------------------
## --- College admission problem

s.prefs <- matrix(c(1,2,3,
                    1,2,3,
                    1,2,3,
                    2,1,3,
                    2,1,3),
                  byrow = FALSE, ncol = 5, nrow = 3); s.prefs
c.prefs <- matrix(c(1,4,2,3,5,
                    5,2,3,4,1,
                    1,2,3,4,5),
                  byrow = FALSE, ncol = 3, nrow = 5); c.prefs
nSlots <- c(2,2,1)

## Boston mechanism
 iaa(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots)$matchings

## Gale-Shapley algorithm
 iaa(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots, acceptance="deferred")$matchings

## Same results for the Gale-Shapley algorithm with hri2() function (but different format)
 set.seed(123)
 iaa(nStudents=7, nSlots=c(3,3), acceptance="deferred")$matchings
 set.seed(123)
 hri2(nStudents=7, nSlots=c(3,3))$matchings
 ##}
##\dontrun{
## --------------------------------
## --- College admission problem

s.prefs <- matrix(c(1,2,3,
                    1,2,3,
                    1,2,3,
                    2,1,3,
                    2,1,3),
                  byrow = FALSE, ncol = 5, nrow = 3); s.prefs
c.prefs <- matrix(c(1,4,2,3,5,
                    5,2,3,4,1,
                    1,2,3,4,5),
                  byrow = FALSE, ncol = 3, nrow = 5); c.prefs
nSlots <- c(2,2,1)

## Boston mechanism
 iaa(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots)$matchings

## Gale-Shapley algorithm
 iaa(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots, acceptance="deferred")$matchings

## Same results for the Gale-Shapley algorithm with hri2() function (but different format)
 set.seed(123)
 iaa(nStudents=7, nSlots=c(3,3), acceptance="deferred")$matchings
 set.seed(123)
 hri2(nStudents=7, nSlots=c(3,3))$matchings
 ##}

Karlson-Holm-Breen method for comparing probit coefficients

Description

Significance test for confounding; that is, the difference between regression coefficients from same-sample nested logit and probit models. The test procedure follows Karlson et al (2012), Section 3.4.

Usage

khb(X, y, z)
khb(X, y, z)

Arguments

`X`	data frame comprising independent variables including confounding variable.
`y`	vector of dependent variable.
`z`	character string giving the name of the confounding variable in `X`.

Author(s)

Thilo Klein

References

Karlson, K.B., A. Holm and R. Breen (2012). Comparing regression coefficients between same-sample nested models using logit and probit: A new method. Sociological Methodology, 42(1):286–313.

Examples

## 1. load results from Klein (2015a)
 data(klein15a)

## 2. apply KHB method
with(klein15a$variables, khb(X=X, y=Y, z="eta"))
## 1. load results from Klein (2015a)
 data(klein15a)

## 2. apply KHB method
with(klein15a$variables, khb(X=X, y=Y, z="eta"))

MCMC results in Klein (2015a)

Description

MCMC results in Klein (2015a).

Usage

data(klein15a)data(klein15a)

Format

A list containing the following elements:

model.list: .
coefs: .

References

Klein, T. (2015a). Does Anti-Diversification Pay? A One-Sided Matching Model of Microcredit. Cambridge Working Papers in Economics, #1521.

Results of Monte Carlo Simulations in Klein (2015b)

Description

Results of Monte Carlo Simulations in Klein (2015b) for 40 two-group markets.

Usage

data(klein15b)data(klein15b)

Format

A list containing the following elements:

exp.5.5.ols: Benchmark study, OLS: coefficient estimates for 40 markets with groups of 5. Data for all 5 group members is observed.
exp.5.5.ntu: Benchmark study, structural model.
exp.6.5.ols: Experiment 1, OLS: coefficient estimates for 40 markets with groups of 6. Only Data for 5 group members is observed.
exp.6.5.ntu: Experiment 1, structural model.
exp.6.6.ols: Experiment 2, OLS: coefficient estimates for 40 markets with groups of 6. Data for all 6 group members is observed but only a random sample of 250 of the 922 counterfactual groups is used in the analysis.
exp.6.6.ntu: Experiment 2, structural model.

References

Klein, T. (2015a). Does Anti-Diversification Pay? A One-Sided Matching Model of Microcredit. Cambridge Working Papers in Economics, #1521.

Klein, T. (2015b). Analysis of stable matchings in R: Package matchingMarkets. Vignette to R package matchingMarkets, The Comprehensive R Archive Network.

Examples

## Plot of posterior distributions

data(klein15b)

tpe <- c(rep("Benchmark",2), rep("Experiment 1",2), rep("Experiment 2",2))

for(i in seq(1,length(klein15b)-1,2)){
  ntu <- klein15b[[i]]
  ols <- klein15b[[i+1]]

  ntu <- ntu[,colnames(ntu) == "beta.wst.ieq"]
  ols <- ols[,colnames(ols) == "beta.wst.ieq"]
  
  if(i == 1){
    draws <- data.frame(Structural=ntu, OLS=ols, type=tpe[i]) #, stringsAsFactors=FALSE
  } else{
    draws <- rbind(draws, data.frame(Structural=ntu, OLS=ols, type=tpe[i]))
  }
}

library(lattice)
lattice.options(default.theme = standard.theme(color = FALSE))
keys <- list(text=c("Structural model","OLS"), space="top", columns=2, lines=TRUE)
densityplot( ~ Structural + OLS | type, plot.points=FALSE, auto.key=keys,
       data = draws, xlab = "coefficient draws", ylab = "density", type = "l",
       panel = function(x,...) {
         panel.densityplot(x,...)
         panel.abline(v=-1, lty=3)
       })

## Not run: 
## Modes of posterior distributions

## load data
data(klein15b)

## define function to obtain the mode
mode <- function(x){
  d <- density(x,bw="SJ")
  formatC(round(d$x[which.max(d$y)], 3), format='f', digits=3)
}

## Benchmark study
apply(klein15b$exp.5.5.ntu, 2, mode)
apply(klein15b$exp.5.5.ols, 2, mode)

## Experiment 1
apply(klein15b$exp.6.5.ntu, 2, mode)
apply(klein15b$exp.6.5.ols, 2, mode)

## Experiment 2
apply(klein15b$exp.6.6.ntu, 2, mode)
apply(klein15b$exp.6.6.ols, 2, mode)

## End(Not run)
## Plot of posterior distributions

data(klein15b)

tpe <- c(rep("Benchmark",2), rep("Experiment 1",2), rep("Experiment 2",2))

for(i in seq(1,length(klein15b)-1,2)){
  ntu <- klein15b[[i]]
  ols <- klein15b[[i+1]]

  ntu <- ntu[,colnames(ntu) == "beta.wst.ieq"]
  ols <- ols[,colnames(ols) == "beta.wst.ieq"]
  
  if(i == 1){
    draws <- data.frame(Structural=ntu, OLS=ols, type=tpe[i]) #, stringsAsFactors=FALSE
  } else{
    draws <- rbind(draws, data.frame(Structural=ntu, OLS=ols, type=tpe[i]))
  }
}

library(lattice)
lattice.options(default.theme = standard.theme(color = FALSE))
keys <- list(text=c("Structural model","OLS"), space="top", columns=2, lines=TRUE)
densityplot( ~ Structural + OLS | type, plot.points=FALSE, auto.key=keys,
       data = draws, xlab = "coefficient draws", ylab = "density", type = "l",
       panel = function(x,...) {
         panel.densityplot(x,...)
         panel.abline(v=-1, lty=3)
       })

## Not run: 
## Modes of posterior distributions

## load data
data(klein15b)

## define function to obtain the mode
mode <- function(x){
  d <- density(x,bw="SJ")
  formatC(round(d$x[which.max(d$y)], 3), format='f', digits=3)
}

## Benchmark study
apply(klein15b$exp.5.5.ntu, 2, mode)
apply(klein15b$exp.5.5.ols, 2, mode)

## Experiment 1
apply(klein15b$exp.6.5.ntu, 2, mode)
apply(klein15b$exp.6.5.ols, 2, mode)

## Experiment 2
apply(klein15b$exp.6.6.ntu, 2, mode)
apply(klein15b$exp.6.6.ols, 2, mode)

## End(Not run)

MC Experiments

Description

MC Experiments

Usage

mce(seed, niter, N, n, m, type, method)
mce(seed, niter, N, n, m, type, method)

Arguments

`seed`	seed value for Monte Carlo Experiment
`niter`	number of draws in estimation
`N`	group size (population)
`n`	group size (sample)
`m`	number of markets
`type`	type of the MC Experiment. Either `group.members` for randomly sampled group members or `counterfactual.groups` for randomly sampled number of counterfactual (or feasible) groups in selection equation (capped at limit max.combs=250)
`method`	either `group.members` or `counterfactual.groups`

Author(s)

Thilo Klein

Examples

## Not run: 
## 1. Set parameters
mciter <- 2 #500
niter <- 10 #400000
nodes <- 4

## 2. Setup parallel backend to use 4 processors
library(foreach); library(doSNOW)
cl <- makeCluster(4); registerDoSNOW(cl)

## 3. Define foreach loop function
mce.add <- function(mciter, niter, N, n, m, type, method){
  h <- foreach(i=1:mciter) %dopar% {
    library(matchingMarkets)
    mce(seed=i,niter, N, n, m, type, method)
  }
  do.call(rbind, h)
}

## 4. Run siumlations:

## 4-a. Benchmark study
exp.5.5.ols <- mce.add(mciter=mciter, niter=niter, N=5, n=5, m=40,
                       type="group.members", method="outcome")
exp.5.5.ntu <- mce.add(mciter=mciter, niter=niter, N=5, n=5, m=40, 
                       type="group.members", method="NTU")

## 4-b. Experiment 1: randomly sampled group members
exp.6.5.ols <- mce.add(mciter=mciter, niter=niter, N=6, n=5, m=40,
                       type="group.members", method="outcome")
exp.6.5.ntu <- mce.add(mciter=mciter, niter=niter, N=6, n=5, m=40,
                       type="group.members", method="NTU")

## 4-c. Experiment 2: randomly sampled counterfactual groups
exp.6.6.ols <- mce.add(mciter=mciter, niter=niter, N=6, n=6, m=40, 
                       type="counterfactual.groups", method="outcome")
exp.6.6.ntu <- mce.add(mciter=mciter, niter=niter, N=6, n=6, m=40, 
                       type="counterfactual.groups", method="NTU")

## 5. Stop parallel backend
stopCluster(cl)

## End(Not run)
## Not run: 
## 1. Set parameters
mciter <- 2 #500
niter <- 10 #400000
nodes <- 4

## 2. Setup parallel backend to use 4 processors
library(foreach); library(doSNOW)
cl <- makeCluster(4); registerDoSNOW(cl)

## 3. Define foreach loop function
mce.add <- function(mciter, niter, N, n, m, type, method){
  h <- foreach(i=1:mciter) %dopar% {
    library(matchingMarkets)
    mce(seed=i,niter, N, n, m, type, method)
  }
  do.call(rbind, h)
}

## 4. Run siumlations:

## 4-a. Benchmark study
exp.5.5.ols <- mce.add(mciter=mciter, niter=niter, N=5, n=5, m=40,
                       type="group.members", method="outcome")
exp.5.5.ntu <- mce.add(mciter=mciter, niter=niter, N=5, n=5, m=40, 
                       type="group.members", method="NTU")

## 4-b. Experiment 1: randomly sampled group members
exp.6.5.ols <- mce.add(mciter=mciter, niter=niter, N=6, n=5, m=40,
                       type="group.members", method="outcome")
exp.6.5.ntu <- mce.add(mciter=mciter, niter=niter, N=6, n=5, m=40,
                       type="group.members", method="NTU")

## 4-c. Experiment 2: randomly sampled counterfactual groups
exp.6.6.ols <- mce.add(mciter=mciter, niter=niter, N=6, n=6, m=40, 
                       type="counterfactual.groups", method="outcome")
exp.6.6.ntu <- mce.add(mciter=mciter, niter=niter, N=6, n=6, m=40, 
                       type="counterfactual.groups", method="NTU")

## 5. Stop parallel backend
stopCluster(cl)

## End(Not run)

Partitioning Linear Programme for the stable roommates problem

Description

Finds the stable matching in the stable roommates problem with transferable utility. Uses the Partitioning Linear Programme formulated in Quint (1991).

Usage

plp(V = NULL, N = NULL)
plp(V = NULL, N = NULL)

Arguments

`V`	valuation matrix of dimension `NxN` that gives row-players valuation over column players (or vice versa).
`N`	integer (divisible by 2) that gives the number of players in the market.

Value

plp returns a list with the following items.

`Valuation.matrix`	input values of V.
`Assignment.matrix`	upper triangular matrix of dimension `NxN` with entries of 1 for equilibrium pairs and 0 otherwise.
`Equilibrium.groups`	matrix that gives the `N/2` equilibrium pairs and equilibrium partners' mutual valuations.

Author(s)

Thilo Klein

References

Quint, T. (1991). Necessary and sufficient conditions for balancedness in partitioning games. Mathematical Social Sciences, 22(1):87–91.

Examples

## Roommate problem with 10 players, transferable utility and random preferences:
plp(N=10)

## Roommate problem with 10 players, transferable utility and given preferences:
V <- matrix(rep(1:10, 10), 10, 10)
plp(V=V)
## Roommate problem with 10 players, transferable utility and random preferences:
plp(N=10)

## Roommate problem with 10 players, transferable utility and given preferences:
V <- matrix(rep(1:10, 10), 10, 10)
plp(V=V)

Predict method for fitted matching models

Description

Calculate predicted values for matching models fitted with functions stabit and stabit2.

Usage

## S3 method for class 'stabit2'
predict(object, newdata = NULL, ...)
## S3 method for class 'stabit2'
predict(object, newdata = NULL, ...)

Arguments

`object`	a fitted object of class `stabit`
`newdata`	optionally, a data frame in which to look for variables with which to predict. If omitted, the fitted linear predictors or the fitted response values are returned.
`...`	.

Author(s)

Thilo Klein

References

Klein, T. (2015a). Does Anti-Diversification Pay? A One-Sided Matching Model of Microcredit. Cambridge Working Papers in Economics, #1521.

Examples


## load the results from Klein (2015) paper
 data(klein15a)
 
## predict the latent outcome variable
 predict(klein15a)

## load the results from Klein (2015) paper
 data(klein15a)
 
## predict the latent outcome variable
 predict(klein15a)

Random serial dictatorship mechanism

Description

Implements the random serial dictatorship algorithm algorithm for a fair division of indivisible objects among individuals. The mechanism takes individuals' prioirty order as an input or alternatively draws a random permutation of the agents form the uniform distribution. Individuals are then successively assigned an object in that order (so the first agent in the ordering gets the first pick and so on).

Usage

rsd(
  nIndividuals = ncol(prefs),
  nObjects = nrow(prefs),
  prefs,
  priority,
  seed = NULL,
  nSlots = rep(1, nObjects)
)
rsd(
  nIndividuals = ncol(prefs),
  nObjects = nrow(prefs),
  prefs,
  priority,
  seed = NULL,
  nSlots = rep(1, nObjects)
)

Arguments

`nIndividuals`	integer indicating the number of individuals in the matching problem. Defaults to ncol(prefs).
`nObjects`	integer indication the number of objects in the matching problem. Defaults to nrow(prefs).
`prefs`	matrix of dimension `nObjects` x `nIndividuals` with the jth column containing students j's ranking over the objects in decreasing order of preference (i.e. most preferred first).
`priority`	(Optional) vector of length `nIndividuals` indicating the priority of the individuals in decreasing order (i.e. highest individuals first). If none is given, a random order is chosen.
`seed`	(Optional) integer setting the state for random number generation. Defaults to seed = 123.
`nSlots`	(Optional) vector of length `nObjects` indicating the owners/slots possible/available for each object. (If nSlots only consists of ones this means that every object can only have one owner, if the numbers are higher interpreting objects for example as schools might be more intuitive). Defaults to a vector of length `nObjects` filled with ones.

Value

rsd returns a data frame containing the final matching of individuals (ind) to objects (obj).

Author(s)

Thilo Klein, Alexander Sauer

Examples

## Generate preference-matrix
prefs <- matrix(c(1,2,3,
                  3,1,2,
                  1,3,2),
                  byrow = FALSE, ncol = 3)

priority <- c(1,2,3)
nSlots <- c(1,1,1)

rsd(prefs = prefs, priority = priority, nSlots = nSlots)
## Generate preference-matrix
prefs <- matrix(c(1,2,3,
                  3,1,2,
                  1,3,2),
                  byrow = FALSE, ncol = 3)

priority <- c(1,2,3)
nSlots <- c(1,1,1)

rsd(prefs = prefs, priority = priority, nSlots = nSlots)

All stable matchings in the stable roommates problem with incomplete lists

Description

Finds all stable matchings (if one exists) in the stable roommates problem with incomplete lists using the Prosser (2014) constraint encoding based on either given or randomly generated preferences.

Usage

sri(prefs = NULL, nAgents = NULL, seed = NULL, p.range = NULL)
sri(prefs = NULL, nAgents = NULL, seed = NULL, p.range = NULL)

Arguments

`prefs`	valuation matrix of dimension `nAgents x nAgents` that gives column-players' ranking over other players in decreasing order of preference (i.e. most preferred first).
`nAgents`	integer that gives the number of players in the market.
`seed`	integer setting the state for random number generation.
`p.range`	range of two intergers `p.range = c(p.min, p.max)`, where `p.min < p.max`. Produces incomplete preference lists with the length of each player's list randomly sampled from the range `[p.min, p.max]`. Note: interval is always too small because non-permissible matches are automatically deleted.

Value

sri returns a list with the following items.

`prefs`	agents' preference list.
`matching`	edgelist of matched pairs, inculding the number of the match (`matching`).

Author(s)

Thilo Klein

References

Gusfield, D.M. and R.W. Irving (1989). The Stable Marriage Problem: Structure and Algorithms, MIT Press.

Prosser, P. (2014). Stable Roommates and Constraint Programming. Lecture Notes in Computer Science, CPAIOR 2014 Edition. Springer International Publishing, 8451: 15–28.

Irving, R.W. and S. Scott (2007). The stable fixtures problem: A many-to-many extension of stable roommates. Discrete Applied Mathematics, 155: 2118–2129.

Examples

## Roommate problem with 10 players, given preferences:
 prefs <- matrix(rep(1:10, 10), 10, 10)
 sri(prefs=prefs)

## Roommate problem with 10 players, random preferences:
 sri(nAgents=10, seed=1)

## Roommate problem with no equilibrium matching:
 sri(nAgents=10, seed=2)

## Roommate problem with 3 equilibria:
 sri(nAgents=10, seed=3)

## Roommate problem with 10 players, given preferences:
 prefs <- matrix(rep(1:10, 10), 10, 10)
 sri(prefs=prefs)

## Roommate problem with 10 players, random preferences:
 sri(nAgents=10, seed=1)

## Roommate problem with no equilibrium matching:
 sri(nAgents=10, seed=2)

## Roommate problem with 3 equilibria:
 sri(nAgents=10, seed=3)

Stability-Check

Description

Checks a given two sided matching for blocking pairs.

Usage

stabchk(
  matching,
  c.prefs,
  s.prefs,
  nColleges = ncol(c.prefs),
  nStudents = ncol(s.prefs)
)
stabchk(
  matching,
  c.prefs,
  s.prefs,
  nColleges = ncol(c.prefs),
  nStudents = ncol(s.prefs)
)

Arguments

`matching`	data frame or matrix of dimension (`min[nColleges, nStudents]`) x 2 containing in column 1 the colleges and in column 2 the students with each row forming a couple.
`c.prefs`	matrix of dimension `nStudents` x `nColleges` with column j containing college j'th ranking over students in decreasing order of preferences.
`s.prefs`	matrix of dimension `nColleges` x `nStudents` with column j containing student j'th ranking over colleges in decreasing order of preferences.
`nColleges`	integer indicating the number of colleges
`nStudents`	integer indicating the number of students

Value

stabchk returns a data frame with as many rows as blocking pairs were found. Column 1 indicates the college and column 2 indicate the student of the blocking pairs. Returns NULL if no blocking pair is found.

Author(s)

Thilo Klein, Alexander Sauer

Examples



## 1-a. Generate preferences for colleges
c.prefs = matrix(c(1,2,3,
                   3,2,1,
                   3,2,1),
                    byrow = FALSE, ncol = 3); c.prefs

## 1-b. Generate preferences for students
s.prefs = matrix(c(1,2,3,
                   3,2,1,
                   2,1,3),
                 byrow = FALSE, ncol = 3);s.prefs

## 1-c. Generate matching
matching = matrix(c(1,2,
                    2,1,
                    3,3),
                  byrow = TRUE, ncol = 2); matching

## 1-d. Check stability
stabchk(matching = matching, c.prefs = c.prefs, s.prefs = s.prefs)

## 2-a. Generate new matching without blocking pairs as a data frame
matching = data.frame('colleges' = c(1,2,3), 'student' = c(1,3,2))
stabchk(matching = matching, c.prefs = c.prefs, s.prefs = s.prefs)

## 3-a. Example with missing values:
matching  <- matrix(c(1,1,2,2,3,3), byrow = FALSE, ncol = 2)
c.prefs <- matrix(c(1,1,3,rep(NA, 6)), byrow = TRUE, ncol = 3)
s.prefs <- matrix(c(2,2,3,rep(NA, 6)), byrow = TRUE, ncol = 3)
stabchk(matching = matching, c.prefs = c.prefs, s.prefs = s.prefs)
## 1-a. Generate preferences for colleges
c.prefs = matrix(c(1,2,3,
                   3,2,1,
                   3,2,1),
                    byrow = FALSE, ncol = 3); c.prefs

## 1-b. Generate preferences for students
s.prefs = matrix(c(1,2,3,
                   3,2,1,
                   2,1,3),
                 byrow = FALSE, ncol = 3);s.prefs

## 1-c. Generate matching
matching = matrix(c(1,2,
                    2,1,
                    3,3),
                  byrow = TRUE, ncol = 2); matching

## 1-d. Check stability
stabchk(matching = matching, c.prefs = c.prefs, s.prefs = s.prefs)

## 2-a. Generate new matching without blocking pairs as a data frame
matching = data.frame('colleges' = c(1,2,3), 'student' = c(1,3,2))
stabchk(matching = matching, c.prefs = c.prefs, s.prefs = s.prefs)

## 3-a. Example with missing values:
matching  <- matrix(c(1,1,2,2,3,3), byrow = FALSE, ncol = 2)
c.prefs <- matrix(c(1,1,3,rep(NA, 6)), byrow = TRUE, ncol = 3)
s.prefs <- matrix(c(2,2,3,rep(NA, 6)), byrow = TRUE, ncol = 3)
stabchk(matching = matching, c.prefs = c.prefs, s.prefs = s.prefs)

Matching model and selection correction for group formation

Description

The function provides a Gibbs sampler for a structural matching model that estimates preferences and corrects for sample selection bias when the selection process is a one-sided matching game; that is, group/coalition formation.

The input is individual-level data of all group members from one-sided matching marktes; that is, from group/coalition formation games.

In a first step, the function generates a model matrix with characteristics of all feasible groups of the same size as the observed groups in the market.

For example, in the stable roommates problem with $n=4$ students $\{1,2,3,4\}$ sorting into groups of 2, we have ${4 \choose 2}=6$ feasible groups: (1,2)(3,4) (1,3)(2,4) (1,4)(2,3).

In the group formation problem with $n=6$ students $\{1,2,3,4,5,6\}$ sorting into groups of 3, we have ${6 \choose 3} =20$ feasible groups. For the same students sorting into groups of sizes 2 and 4, we have ${6 \choose 2} + {6 \choose 4}=30$ feasible groups.

The structural model consists of a selection and an outcome equation. The Selection Equation determines which matches are observed ( $D=1$ ) and which are not ( $D=0$ ).

$\begin{array}{lcl} D &= & 1[V \in \Gamma] \\ V &= & W\alpha + \eta \end{array}$

Here, $V$ is a vector of latent valuations of all feasible matches, ie observed and unobserved, and $1[.]$ is the Iverson bracket. A match is observed if its match valuation is in the set of valuations $\Gamma$ that satisfy the equilibrium condition (see Klein, 2015a). This condition differs for matching games with transferable and non-transferable utility and can be specified using the method argument. The match valuation $V$ is a linear function of $W$ , a matrix of characteristics for all feasible groups, and $\eta$ , a vector of random errors. $\alpha$ is a paramter vector to be estimated.

The Outcome Equation determines the outcome for observed matches. The dependent variable can either be continuous or binary, dependent on the value of the binary argument. In the binary case, the dependent variable $R$ is determined by a threshold rule for the latent variable $Y$ .

$\begin{array}{lcl} R &= & 1[Y > c] \\ Y &= & X\beta + \epsilon \end{array}$

Here, $Y$ is a linear function of $X$ , a matrix of characteristics for observed matches, and $\epsilon$ , a vector of random errors. $\beta$ is a paramter vector to be estimated.

The structural model imposes a linear relationship between the error terms of both equations as $\epsilon = \delta\eta + \xi$ , where $\xi$ is a vector of random errors and $\delta$ is the covariance paramter to be estimated. If $\delta$ were zero, the marginal distributions of $\epsilon$ and $\eta$ would be independent and the selection problem would vanish. That is, the observed outcomes would be a random sample from the population of interest.

Usage

stabit(
  x,
  m.id = "m.id",
  g.id = "g.id",
  R = "R",
  selection = NULL,
  outcome = NULL,
  simulation = "none",
  seed = 123,
  max.combs = Inf,
  method = "NTU",
  binary = FALSE,
  offsetOut = 0,
  offsetSel = 0,
  marketFE = FALSE,
  censored = 0,
  gPrior = FALSE,
  dropOnes = FALSE,
  interOut = 0,
  interSel = 0,
  standardize = 0,
  niter = 10,
  verbose = FALSE
)
stabit(
  x,
  m.id = "m.id",
  g.id = "g.id",
  R = "R",
  selection = NULL,
  outcome = NULL,
  simulation = "none",
  seed = 123,
  max.combs = Inf,
  method = "NTU",
  binary = FALSE,
  offsetOut = 0,
  offsetSel = 0,
  marketFE = FALSE,
  censored = 0,
  gPrior = FALSE,
  dropOnes = FALSE,
  interOut = 0,
  interSel = 0,
  standardize = 0,
  niter = 10,
  verbose = FALSE
)

Arguments

`x`	data frame with individual-level characteristics of all group members including market- and group-identifiers.
`m.id`	character string giving the name of the market identifier variable. Defaults to `"m.id"`.
`g.id`	character string giving the name of the group identifier variable. Defaults to `"g.id"`.
`R`	dependent variable in outcome equation. Defaults to `"R"`.
`selection`	list containing variables and pertaining operators in the selection equation. The format is `operation = "variable"`. See the Details and Examples sections.
`outcome`	list containing variables and pertaining operators in the outcome equation. The format is `operation = "variable"`. See the Details and Examples sections.
`simulation`	should the values of dependent variables in selection and outcome equations be simulated? Options are `"none"` for no simulation, `"NTU"` for non-transferable utility matching, `"TU"` for transferable utility or `"random"` for random matching of individuals to groups. Simulation settings are (i) all model coefficients set to `alpha=beta=1`; (ii) covariance between error terms `delta=0.5`; (iii) error terms `eta` and `xi` are draws from a standard normal distribution.
`seed`	integer setting the state for random number generation if `simulation=TRUE`.
`max.combs`	integer (divisible by two) giving the maximum number of feasible groups to be used for generating group-level characteristics.
`method`	estimation method to be used. Either `"NTU"` or `"TU"` for selection correction using non-transferable or transferable utility matching as selection rule; `"outcome"` for estimation of the outcome equation only; or `"model.frame"` for no estimation.
`binary`	logical: if `TRUE` outcome variable is taken to be binary; if `FALSE` outcome variable is taken to be continuous.
`offsetOut`	vector of integers indicating the indices of columns in `X` for which coefficients should be forced to 1. Use 0 for none.
`offsetSel`	vector of integers indicating the indices of columns in `W` for which coefficients should be forced to 1. Use 0 for none.
`marketFE`	logical: if `TRUE` market-level fixed effects are used in outcome equation; if `FALSE` no market fixed effects are used.
`censored`	draws of the `delta` parameter that estimates the covariation between the error terms in selection and outcome equation are 0:not censored, 1:censored from below, 2:censored from above.
`gPrior`	logical: if `TRUE` the g-prior (Zellner, 1986) is used for the variance-covariance matrix.
`dropOnes`	logical: if `TRUE` one-group-markets are exluded from estimation.
`interOut`	two-colum matrix indicating the indices of columns in `X` that should be interacted in estimation. Use 0 for none.
`interSel`	two-colum matrix indicating the indices of columns in `W` that should be interacted in estimation. Use 0 for none.
`standardize`	numeric: if `standardize>0` the independent variables will be standardized by dividing by `standardize` times their standard deviation. Defaults to no standardization `standardize=0`.
`niter`	number of iterations to use for the Gibbs sampler.
`verbose`	.

Details

Operators for variable transformations in selection and outcome arguments.

add: sum over all group members and divide by group size.
int: sum over all possible two-way interactions $x*y$ of group members and divide by the number of those, given by choose(n,2).
ieq: sum over all possible two-way equality assertions $1[x=y]$ and divide by the number of those.
ive: sum over all possible two-way interactions of vectors of variables of group members and divide by number of those.
inv: ...
dst: sum over all possible two-way distances between players and divide by number of those, where distance is defined as $e^{-|x-y|}$ .

Author(s)

Thilo Klein

References

Klein, T. (2015a). Does Anti-Diversification Pay? A One-Sided Matching Model of Microcredit. Cambridge Working Papers in Economics, #1521.

Zellner, A. (1986). On assessing prior distributions and Bayesian regression analysis with g-prior distributions, volume 6, pages 233–243. North-Holland, Amsterdam.

Examples

## Not run: 
## --- SIMULATED EXAMPLE ---

## 1. Simulate one-sided matching data for 200 markets (m=200) with 2 groups
##    per market (gpm=2) and 5 individuals per group (ind=5). True parameters 
##    in selection equation is wst=1, in outcome equation wst=0. 

## 1-a. Simulate individual-level, independent variables
 idata <- stabsim(m=200, ind=5, seed=123, gpm=2)
 head(idata)
 
## 1-b. Simulate group-level variables 
 mdata <- stabit(x=idata, simulation="NTU", method="model.frame",
 selection = list(add="wst"), outcome = list(add="wst"), verbose=FALSE)
 head(mdata$OUT)
 head(mdata$SEL)


## 2. Bias from sorting

## 2-a. Naive OLS estimation
 lm(R ~ wst.add, data=mdata$OUT)$coefficients

## 2-b. epsilon is correlated with independent variables
 with(mdata$OUT, cor(epsilon, wst.add))
 
## 2-c. but xi is uncorrelated with independent variables
 with(mdata$OUT, cor(xi, wst.add))

## 3. Correction of sorting bias when valuations V are observed

## 3-a. 1st stage: obtain fitted value for eta
lm.sel <- lm(V ~ -1 + wst.add, data=mdata$SEL)
lm.sel$coefficients

eta <- lm.sel$resid[mdata$SEL$D==1]

## 3-b. 2nd stage: control for eta
 lm(R ~ wst.add + eta, data=mdata$OUT)$coefficients


## 4. Run Gibbs sampler
 fit1 <- stabit(x=idata, method="NTU", simulation="NTU", censored=1, 
                selection = list(add="wst"), outcome = list(add="wst"), 
                niter=2000, verbose=FALSE)


## 5. Coefficient table
 summary(fit1)


## 6. Plot MCMC draws for coefficients
 plot(fit1)

## --- REPLICATION, Klein (2015a) ---

## 1. Load data 
 data(baac00); head(baac00)
 
## 2. Run Gibbs sampler
 klein15a <- stabit(x=baac00, selection = list(inv="pi",ieq="wst"), 
        outcome = list(add="pi",inv="pi",ieq="wst",
        add=c("loan_size","loan_size2","lngroup_agei")), offsetOut=1,
        method="NTU", binary=TRUE, gPrior=TRUE, marketFE=TRUE, niter=800000)

## 3. Marginal effects
 summary(klein15a, mfx=TRUE)
 
## 4. Plot MCMC draws for coefficients
 plot(klein15a)

## End(Not run)
## Not run: 
## --- SIMULATED EXAMPLE ---

## 1. Simulate one-sided matching data for 200 markets (m=200) with 2 groups
##    per market (gpm=2) and 5 individuals per group (ind=5). True parameters 
##    in selection equation is wst=1, in outcome equation wst=0. 

## 1-a. Simulate individual-level, independent variables
 idata <- stabsim(m=200, ind=5, seed=123, gpm=2)
 head(idata)
 
## 1-b. Simulate group-level variables 
 mdata <- stabit(x=idata, simulation="NTU", method="model.frame",
 selection = list(add="wst"), outcome = list(add="wst"), verbose=FALSE)
 head(mdata$OUT)
 head(mdata$SEL)


## 2. Bias from sorting

## 2-a. Naive OLS estimation
 lm(R ~ wst.add, data=mdata$OUT)$coefficients

## 2-b. epsilon is correlated with independent variables
 with(mdata$OUT, cor(epsilon, wst.add))
 
## 2-c. but xi is uncorrelated with independent variables
 with(mdata$OUT, cor(xi, wst.add))

## 3. Correction of sorting bias when valuations V are observed

## 3-a. 1st stage: obtain fitted value for eta
lm.sel <- lm(V ~ -1 + wst.add, data=mdata$SEL)
lm.sel$coefficients

eta <- lm.sel$resid[mdata$SEL$D==1]

## 3-b. 2nd stage: control for eta
 lm(R ~ wst.add + eta, data=mdata$OUT)$coefficients


## 4. Run Gibbs sampler
 fit1 <- stabit(x=idata, method="NTU", simulation="NTU", censored=1, 
                selection = list(add="wst"), outcome = list(add="wst"), 
                niter=2000, verbose=FALSE)


## 5. Coefficient table
 summary(fit1)


## 6. Plot MCMC draws for coefficients
 plot(fit1)

## --- REPLICATION, Klein (2015a) ---

## 1. Load data 
 data(baac00); head(baac00)
 
## 2. Run Gibbs sampler
 klein15a <- stabit(x=baac00, selection = list(inv="pi",ieq="wst"), 
        outcome = list(add="pi",inv="pi",ieq="wst",
        add=c("loan_size","loan_size2","lngroup_agei")), offsetOut=1,
        method="NTU", binary=TRUE, gPrior=TRUE, marketFE=TRUE, niter=800000)

## 3. Marginal effects
 summary(klein15a, mfx=TRUE)
 
## 4. Plot MCMC draws for coefficients
 plot(klein15a)

## End(Not run)

Matching model and selection correction for college admissions

Description

The function provides a Gibbs sampler for a structural matching model that estimates preferences and corrects for sample selection bias when the selection process is a two-sided matching game; i.e., a matching of students to colleges.

The structural model consists of a selection and an outcome equation. The Selection Equation determines which matches are observed ( $D=1$ ) and which are not ( $D=0$ ).

$\begin{array}{lcl} D &= & 1[V \in \Gamma] \\ V &= & W\beta + \eta \end{array}$

Here, $V$ is a vector of latent valuations of all feasible matches, ie observed and unobserved, and $1[.]$ is the Iverson bracket. A match is observed if its match valuation is in the set of valuations $\Gamma$ that satisfy the equilibrium condition (see Sorensen, 2007). The match valuation $V$ is a linear function of $W$ , a matrix of characteristics for all feasible matches, and $\eta$ , a vector of random errors. $\beta$ is a paramter vector to be estimated.

$\begin{array}{lcl} R &= & 1[Y > c] \\ Y &= & X\alpha + \epsilon \end{array}$

Here, $Y$ is a linear function of $X$ , a matrix of characteristics for observed matches, and $\epsilon$ , a vector of random errors. $\alpha$ is a paramter vector to be estimated.

The structural model imposes a linear relationship between the error terms of both equations as $\epsilon = \kappa\eta + \nu$ , where $\nu$ is a vector of random errors and $\kappa$ is the covariance paramter to be estimated. If $\kappa$ were zero, the marginal distributions of $\epsilon$ and $\eta$ would be independent and the selection problem would vanish. That is, the observed outcomes would be a random sample from the population of interest.

Usage

stabit2(
  OUT = NULL,
  SEL = NULL,
  colleges = NULL,
  students = NULL,
  outcome = NULL,
  selection,
  binary = FALSE,
  niter,
  gPrior = FALSE,
  censored = 1,
  thin = 1,
  nCores = max(1, detectCores() - 1),
  ...
)
stabit2(
  OUT = NULL,
  SEL = NULL,
  colleges = NULL,
  students = NULL,
  outcome = NULL,
  selection,
  binary = FALSE,
  niter,
  gPrior = FALSE,
  censored = 1,
  thin = 1,
  nCores = max(1, detectCores() - 1),
  ...
)

Arguments

`OUT`	data frame with characteristics of all observed matches, including market identifier `m.id`, college identifier `c.id` and student identifier `s.id`.
`SEL`	optional: data frame with characteristics of all observed and unobserved matches, including market identifier `m.id`, college identifier `c.id` and student identifier `s.id`.
`colleges`	character vector of variable names for college characteristics. These variables carry the same value for any college.
`students`	character vector of variable names for student characteristics. These variables carry the same value for any student.
`outcome`	formula for match outcomes.
`selection`	formula for match valuations.
`binary`	logical: if `TRUE` outcome variable is taken to be binary; if `FALSE` outcome variable is taken to be continuous.
`niter`	number of iterations to use for the Gibbs sampler.
`gPrior`	logical: if `TRUE` the g-prior (Zellner, 1986) is used for the variance-covariance matrix. (Not yet implemented)
`censored`	draws of the `kappa` parameter that estimates the covariation between the error terms in selection and outcome equation are 0:not censored, 1:censored from below, 2:censored from above.
`thin`	integer indicating the level of thinning in the MCMC draws. The default `thin=1` saves every draw, `thin=2` every second, etc.
`nCores`	number of cores to be used in parallel Gibbs sampling.
`...`	.

Author(s)

Thilo Klein

References

Sorensen, M. (2007). How Smart is Smart Money? A Two-Sided Matching Model of Venture Capital. Journal of Finance, 62 (6): 2725-2762.

Examples

## Not run: 
## --- SIMULATED EXAMPLE ---

## 1. Simulate two-sided matching data for 20 markets (m=20) with 100 students
##    (nStudents=100) per market and 20 colleges with quotas of 5 students, each
##    (nSlots=rep(5,20)). True parameters in selection and outcome equations are 
##    all equal to 1.

xdata <- stabsim2(m=20, nStudents=100, nSlots=rep(5,20), verbose=FALSE,
  colleges = "c1", students = "s1",
  outcome = ~ c1:s1 + eta + nu,
  selection = ~ -1 + c1:s1 + eta
)
head(xdata$OUT)


## 2. Correction for sorting bias when match valuations V are observed

## 2-a. Bias from sorting
 lm1 <- lm(y ~ c1:s1, data=xdata$OUT)
 summary(lm1)

## 2-b. Cause of the bias
 with(xdata$OUT, cor(c1*s1, eta))

## 2-c. Correction for sorting bias
 lm2a <- lm(V ~ -1 + c1:s1, data=xdata$SEL); summary(lm2a)
 etahat <- lm2a$residuals[xdata$SEL$D==1]
 
 lm2b <- lm(y ~ c1:s1 + etahat, data=xdata$OUT)
 summary(lm2b)


## 3. Correction for sorting bias when match valuations V are unobserved

## 3-a. Run Gibbs sampler (when SEL is given)
 fit2 <- stabit2(OUT = xdata$OUT, 
           SEL = xdata$SEL,
           outcome = y ~ c1:s1, 
           selection = ~ -1 + c1:s1,
           niter=1000
 )

## 3-b. Alternatively: Run Gibbs sampler (when SEL is not given)
 fit2 <- stabit2(OUT = xdata$OUT, 
           colleges = "c1",
           students = "s1",
           outcome = y ~ c1:s1, 
           selection = ~ -1 + c1:s1,
           niter=1000
 )


## 4. Implemented methods

## 4-a. Get coefficients
 fit2
 
## 4-b. Coefficient table
 summary(fit2)
 
## 4-c. Get marginal effects
 summary(fit2, mfx=TRUE)
 
## 4-d. Also try the following functions
 #coef(fit2)
 #fitted(fit2)
 #residuals(fit2)
 #predict(fit2, newdata=NULL)

   
## 5. Plot MCMC draws for coefficients
 plot(fit2)

## End(Not run)
## Not run: 
## --- SIMULATED EXAMPLE ---

## 1. Simulate two-sided matching data for 20 markets (m=20) with 100 students
##    (nStudents=100) per market and 20 colleges with quotas of 5 students, each
##    (nSlots=rep(5,20)). True parameters in selection and outcome equations are 
##    all equal to 1.

xdata <- stabsim2(m=20, nStudents=100, nSlots=rep(5,20), verbose=FALSE,
  colleges = "c1", students = "s1",
  outcome = ~ c1:s1 + eta + nu,
  selection = ~ -1 + c1:s1 + eta
)
head(xdata$OUT)


## 2. Correction for sorting bias when match valuations V are observed

## 2-a. Bias from sorting
 lm1 <- lm(y ~ c1:s1, data=xdata$OUT)
 summary(lm1)

## 2-b. Cause of the bias
 with(xdata$OUT, cor(c1*s1, eta))

## 2-c. Correction for sorting bias
 lm2a <- lm(V ~ -1 + c1:s1, data=xdata$SEL); summary(lm2a)
 etahat <- lm2a$residuals[xdata$SEL$D==1]
 
 lm2b <- lm(y ~ c1:s1 + etahat, data=xdata$OUT)
 summary(lm2b)


## 3. Correction for sorting bias when match valuations V are unobserved

## 3-a. Run Gibbs sampler (when SEL is given)
 fit2 <- stabit2(OUT = xdata$OUT, 
           SEL = xdata$SEL,
           outcome = y ~ c1:s1, 
           selection = ~ -1 + c1:s1,
           niter=1000
 )

## 3-b. Alternatively: Run Gibbs sampler (when SEL is not given)
 fit2 <- stabit2(OUT = xdata$OUT, 
           colleges = "c1",
           students = "s1",
           outcome = y ~ c1:s1, 
           selection = ~ -1 + c1:s1,
           niter=1000
 )


## 4. Implemented methods

## 4-a. Get coefficients
 fit2
 
## 4-b. Coefficient table
 summary(fit2)
 
## 4-c. Get marginal effects
 summary(fit2, mfx=TRUE)
 
## 4-d. Also try the following functions
 #coef(fit2)
 #fitted(fit2)
 #residuals(fit2)
 #predict(fit2, newdata=NULL)

   
## 5. Plot MCMC draws for coefficients
 plot(fit2)

## End(Not run)

Simulated data for group formation problem

Description

Simulate individual-level data for one-sided matching markets.

Usage

stabsim(m, ind, seed = 123, singles = NULL, gpm = 2)
stabsim(m, ind, seed = 123, singles = NULL, gpm = 2)

Arguments

`m`	integer indicating the number of markets to be simulated.
`ind`	integer (or vector) indicating the number of individuals per group.
`seed`	integer setting the state for random number generation. Defaults to `set.seed(123)`.
`singles`	integer giving the number of one-group markets.
`gpm`	integer giving the number of groups per market.

Value

stabsim returns a data frame with the randomly generated variables mimicking those in dataset baac00.

`m.id`	categorical: market identifier.
`g.id`	categorical: group identifier.
`wst`	binary: indicator taking the value 1 if last year was worse than the year before; 0 otherwise.
`R`	NA: group outcome is not simulated. It can be obtained using the `simulation` argument in function `stabit`.

Author(s)

Thilo Klein

Examples

## Coalitions [gpm := 2 !]
## Simulate one-sided matching data for 4 markets (m=4) with 2 groups
## per market (gpm=2) and 2 to 4 individuals per group (ind=2:4)
 idata <- stabsim(m=4, ind=2:4, seed=124, singles=2, gpm=2)  

## Rommmates [ind := 2 !]
## Simulate one-sided matching data for 3 markets (m=3) with 3 groups
## per market (gpm=3) and 2 individuals per group (ind=2)
 idata <- stabsim(m=3, ind=2, seed=124, gpm=3)
## Coalitions [gpm := 2 !]
## Simulate one-sided matching data for 4 markets (m=4) with 2 groups
## per market (gpm=2) and 2 to 4 individuals per group (ind=2:4)
 idata <- stabsim(m=4, ind=2:4, seed=124, singles=2, gpm=2)  

## Rommmates [ind := 2 !]
## Simulate one-sided matching data for 3 markets (m=3) with 3 groups
## per market (gpm=3) and 2 individuals per group (ind=2)
 idata <- stabsim(m=3, ind=2, seed=124, gpm=3)

Simulated data for college admissions problem

Description

Simulate data for two-sided matching markets. In the simulation for the Sorensen (2007) model with one selection equation, an equal sharing rule of $\lambda = 0.5$ is used.

Usage

stabsim2(
  m,
  nStudents,
  nColleges = length(nSlots),
  nSlots,
  colleges,
  students,
  outcome,
  selection,
  binary = FALSE,
  seed = 123,
  verbose = TRUE
)
stabsim2(
  m,
  nStudents,
  nColleges = length(nSlots),
  nSlots,
  colleges,
  students,
  outcome,
  selection,
  binary = FALSE,
  seed = 123,
  verbose = TRUE
)

Arguments

`m`	integer indicating the number of markets to be simulated.
`nStudents`	integer indicating the number of students per market.
`nColleges`	integer indicating the number of colleges per market.
`nSlots`	vector of length `nColleges` indicating the number of places at each college, i.e. the college's quota.
`colleges`	character vector of variable names for college characteristics. These variables carry the same value for any college.
`students`	character vector of variable names for student characteristics. These variables carry the same value for any student.
`outcome`	formula for match outcomes.
`selection`	formula for match valuations.
`binary`	logical: if `TRUE` outcome variable is binary; if `FALSE` outcome variable is continuous.
`seed`	integer setting the state for random number generation. Defaults to `set.seed(123)`.
`verbose`	.

Value

stabsim2 returns a list with the following items.

`OUT`
`SEL`
`SELc`
`SELs`

Author(s)

Thilo Klein

Examples


## Simulate two-sided matching data for 2 markets (m=2) with 10 students
## (nStudents=10) per market and 3 colleges (nColleges=3) with quotas of
## 2, 3, and 5 students, respectively.

xdata <- stabsim2(m=2, nStudents=10, nSlots=c(2,3,5), verbose=FALSE,
  colleges = "c1", students = "s1",
  outcome = ~ c1:s1 + eta + nu,
  selection = ~ -1 + c1:s1 + eta
)
head(xdata$OUT)
head(xdata$SEL)

## Simulate two-sided matching data for 2 markets (m=2) with 10 students
## (nStudents=10) per market and 3 colleges (nColleges=3) with quotas of
## 2, 3, and 5 students, respectively.

xdata <- stabsim2(m=2, nStudents=10, nSlots=c(2,3,5), verbose=FALSE,
  colleges = "c1", students = "s1",
  outcome = ~ c1:s1 + eta + nu,
  selection = ~ -1 + c1:s1 + eta
)
head(xdata$OUT)
head(xdata$SEL)

Top-Trading-Cycles Algorithm with existing tenants

Description

Implements an algorithm for the house allocation problem proposed by Abdulkadiroglu and Sonmez (1999) for a matching problem in which there are both vacant houses and existing tenants.

Usage

ttc(
  nStudents = ncol(s.prefs),
  nHouses = length(houses),
  s.prefs,
  houses,
  priority = NULL,
  seed = NULL
)
ttc(
  nStudents = ncol(s.prefs),
  nHouses = length(houses),
  s.prefs,
  houses,
  priority = NULL,
  seed = NULL
)

Arguments

`nStudents`	integer indicating the number of students. Defaults to `ncol(s.prefs)`.
`nHouses`	integer indicating the number of houses. Defaults to `length(houses)`.
`s.prefs`	matrix of dimension `nHouses` x `nStudents` with column j containing student jth ranking over houses in decreasing order of preferences (i.e. most preferred first).
`houses`	vector of length `nHouses` which represents the occupation of the houses. Entry in `k` contains `j` if student `j` is living in house `k` and `NA` if house `k` is vacant.
`priority`	(Optional) vector of length `nStudents`. Gives the prioirity ordering of the students in the search for cycles (Do not confuse it with the preferences!), if nothing is specified a random ordering is chosen.
`seed`	(Optional) integer setting the state for random number generation. Defaults to seed = NULL

Value

ttc returns a data frame of the matching of students (int) to houses (obj) for the house allocation problem based on the Top-Trading-Cycles algorithm.

Author(s)

Thilo Klein, Alexander Sauer

References

Abdulkadiroglu, A. and T. Sonmez (1999). House Allocation with Existing Tenants. Journal of Economic Theory, 88 (2): 233-260.

Shapley, L. and H. Scarf (1974). On Cores and Indivisibility. Journal of Mathematical Economics, 1(1): 23-37.

Examples

##\dontrun{
## 1-a. Generate matrix of individuals' preference rankings over objects,
## a.k.a. Rank Order Lists (ROL).
s.prefs <- matrix(c(3,2,4,1,        # ROL of student 1
                   3,5,6, NA,
                   3,1, NA,NA,
                   2,5,6,4,
                   1,3,2,NA,
                   2,4,5,6), nrow = 4, ncol = 6, byrow = FALSE); s.prefs

## 1-b. Generate vector of house occupation objects ('obj') and their owners ('ind')
(houses <- 1:6)

## 1-c. Find assignment based on TTC algorithm
ttc(s.prefs = s.prefs, houses = houses, nHouses = 6, priority = 1:6)

## 2-a.Compare the example in the paper Abdulkadiroglu et al. (1999)
## on page 246-248 (section 5.1 An Example):
## generate matrix of students' preference rankings over houses, a.k.a. Rank Order Lists (ROL)
s.prefs <- matrix(c(2,6,5,1,4,3,7,NA,
                 7,1,6,5,4,3,2,NA,
                 2,1,4,7,3,6,5,NA,
                 2,4,3,6,1,7,5,NA,
                 4,3,7,1,2,5,6,NA), byrow = FALSE, ncol= 5); s.prefs

## 2-b. Generate house occupation, so student 1 lives in house 1, ..., student 4 lives in house 4
## and the other houses are vacant.
houses <- c(1,2,3,4,NA,NA,NA,NA); houses

## 2-c. Generate priority ordering
priority <- 1:5

## 2-d. Find assigment
ttc(s.prefs = s.prefs, houses = houses, priority = priority)
##}
##\dontrun{
## 1-a. Generate matrix of individuals' preference rankings over objects,
## a.k.a. Rank Order Lists (ROL).
s.prefs <- matrix(c(3,2,4,1,        # ROL of student 1
                   3,5,6, NA,
                   3,1, NA,NA,
                   2,5,6,4,
                   1,3,2,NA,
                   2,4,5,6), nrow = 4, ncol = 6, byrow = FALSE); s.prefs

## 1-b. Generate vector of house occupation objects ('obj') and their owners ('ind')
(houses <- 1:6)

## 1-c. Find assignment based on TTC algorithm
ttc(s.prefs = s.prefs, houses = houses, nHouses = 6, priority = 1:6)

## 2-a.Compare the example in the paper Abdulkadiroglu et al. (1999)
## on page 246-248 (section 5.1 An Example):
## generate matrix of students' preference rankings over houses, a.k.a. Rank Order Lists (ROL)
s.prefs <- matrix(c(2,6,5,1,4,3,7,NA,
                 7,1,6,5,4,3,2,NA,
                 2,1,4,7,3,6,5,NA,
                 2,4,3,6,1,7,5,NA,
                 4,3,7,1,2,5,6,NA), byrow = FALSE, ncol= 5); s.prefs

## 2-b. Generate house occupation, so student 1 lives in house 1, ..., student 4 lives in house 4
## and the other houses are vacant.
houses <- c(1,2,3,4,NA,NA,NA,NA); houses

## 2-c. Generate priority ordering
priority <- 1:5

## 2-d. Find assigment
ttc(s.prefs = s.prefs, houses = houses, priority = priority)
##}

Top-Trading-Cycles Algorithm for a two sided matching problem

Description

Implements the school matching algorithm proposed in Abdulkadiroglu and Sonmez (2003) for a matching problem in which both sides have preferences. Missing preferences are handled in the following ways: Suppose that a student only ranked colleges that are already matched to other students. This student is removed from the matching process and a list with all unmatchable students is printed. If full_return is set to TRUE, a vector with this students is returned as well. Now suppose during the matching process a student points to a college that still has capacities but does not rank any more students. We assume now that the college is indifferent over all other students (so we do not allow for free capacieties) and we match the student who wants to go there to the college.

Usage

ttc2(
  nStudents = ncol(s.prefs),
  nColleges = ncol(c.prefs),
  s.prefs = NULL,
  c.prefs = NULL,
  nSlots = NULL,
  priority = NULL,
  seed = NULL,
  full_return = FALSE
)
ttc2(
  nStudents = ncol(s.prefs),
  nColleges = ncol(c.prefs),
  s.prefs = NULL,
  c.prefs = NULL,
  nSlots = NULL,
  priority = NULL,
  seed = NULL,
  full_return = FALSE
)

Arguments

`nStudents`	integer indicating the number of students in the matching problem. Defaults to `ncol(s.prefs)`.
`nColleges`	integer indicating the number of colleges in the matching problem. Defaults to `ncol(c.prefs)`.
`s.prefs`	matrix of dimension `nColleges` x `nStudents` with the jth column containing student j's ranking over colleges in decreasing order of preference (i.e. most preferred first).
`c.prefs`	matrix of dimension `nStudents` x `nColleges` with the ith column containing college i's ranking over students in decreasing order of preference (i.e. most preferred first).
`nSlots`	vector of length `nColleges` indicating the number of places (i.e. quota) of each college.
`priority`	(Optional) vector of length `nStudents`. Gives the prioirity ordering of the students in the search for cycles (Do not confuse it with the preferences!), if nothing is specified a random ordering is chosen.
`seed`	(Optional) integer setting the state for random number generation. Defaults to seed = NULL
`full_return`	(Optinal) If `TRUE` the return value is a list with the matching, the remaining seats and the unmatchable students is returned. Defaults to `FALSE` and only the matching is returned.

Value

ttc2 returns a data frame of the matching of students (ind) to colleges (obj) for the school market problem based on the Top-Trading-Cycles algorithm.

Author(s)

Thilo Klein, Alexander Sauer

References

Abdulkadiroglu, A. and T. Sonmez (2003). School Choice: A Mechanism Design Approach. American Economic Review, 93 (3): 729-747.

Examples

##\dontrun{
## 1-a. Compare example from the Abdulkadiroglu et al. (2003) (in the Appendix, page 742-744)
## 1-b. Generate matrix of students' preference rankings over schools, a.k.a. Rank Order Lists (ROL)
s.prefs <- matrix(c(
                  2,1,3,4,
                  1,2,3,4,
                  3,2,1,4,
                  3,4,1,2,
                  1,3,4,2,
                  4,1,2,3,
                  1,2,3,4,
                  1,2,4,3),
                  byrow = FALSE, ncol = 8); s.prefs

## 1-c. Generate matrix of schools' preference rankings over students, a.k.a. Rank Order Lists (ROL)
c.prefs <- matrix(c(
                  1,2,3,4,5,6,7,8,
                  3,5,4,8,7,2,1,6,
                  5,3,1,7,2,8,6,4,
                  6,8,7,4,2,3,5,1),
                  byrow = FALSE, ncol = 4); c.prefs

## 1-d. Generate capacities
nSlots <- c(2,2,3,3)

## 1-e. Find assignment based on TTC algorithm
ttc2(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots)

## 2-a. Generate college preferences with college 1 only ranking student 1
c.prefs <- matrix(c(
                   1,rep(NA,7),
                   3,5,4,8,7,2,1,6,
                   5,3,1,7,2,8,6,4,
                   6,8,7,4,2,3,5,1),
                   byrow = FALSE, ncol = 4); c.prefs

## 2-b. Find assignment based on TTC algorithm
ttc2(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots, priority = 1:8)

## If all schools have the same preferences the two sided ttc and the serial dictator yield
## the same outcome if the preferences are taken to be the prioirty order for the serial dictator

# Preferences are the same for all schools:
c.prefs <- matrix(c(
                  5,3,1,7,2,8,6,4,
                  5,3,1,7,2,8,6,4,
                  5,3,1,7,2,8,6,4,
                  5,3,1,7,2,8,6,4),
                  byrow = FALSE, ncol = 4)
priority <- c.prefs[,1]

match_ttc <- ttc2(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots); match_ttc
match_sd <- rsd(prefs = s.prefs, priority = priority, nSlots = nSlots); match_sd
all(match_ttc == match_sd)
##}
##\dontrun{
## 1-a. Compare example from the Abdulkadiroglu et al. (2003) (in the Appendix, page 742-744)
## 1-b. Generate matrix of students' preference rankings over schools, a.k.a. Rank Order Lists (ROL)
s.prefs <- matrix(c(
                  2,1,3,4,
                  1,2,3,4,
                  3,2,1,4,
                  3,4,1,2,
                  1,3,4,2,
                  4,1,2,3,
                  1,2,3,4,
                  1,2,4,3),
                  byrow = FALSE, ncol = 8); s.prefs

## 1-c. Generate matrix of schools' preference rankings over students, a.k.a. Rank Order Lists (ROL)
c.prefs <- matrix(c(
                  1,2,3,4,5,6,7,8,
                  3,5,4,8,7,2,1,6,
                  5,3,1,7,2,8,6,4,
                  6,8,7,4,2,3,5,1),
                  byrow = FALSE, ncol = 4); c.prefs

## 1-d. Generate capacities
nSlots <- c(2,2,3,3)

## 1-e. Find assignment based on TTC algorithm
ttc2(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots)

## 2-a. Generate college preferences with college 1 only ranking student 1
c.prefs <- matrix(c(
                   1,rep(NA,7),
                   3,5,4,8,7,2,1,6,
                   5,3,1,7,2,8,6,4,
                   6,8,7,4,2,3,5,1),
                   byrow = FALSE, ncol = 4); c.prefs

## 2-b. Find assignment based on TTC algorithm
ttc2(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots, priority = 1:8)

## If all schools have the same preferences the two sided ttc and the serial dictator yield
## the same outcome if the preferences are taken to be the prioirty order for the serial dictator

# Preferences are the same for all schools:
c.prefs <- matrix(c(
                  5,3,1,7,2,8,6,4,
                  5,3,1,7,2,8,6,4,
                  5,3,1,7,2,8,6,4,
                  5,3,1,7,2,8,6,4),
                  byrow = FALSE, ncol = 4)
priority <- c.prefs[,1]

match_ttc <- ttc2(s.prefs = s.prefs, c.prefs = c.prefs, nSlots = nSlots); match_ttc
match_sd <- rsd(prefs = s.prefs, priority = priority, nSlots = nSlots); match_sd
all(match_ttc == match_sd)
##}

Top-Trading-Cycles and Chains Algorithm

Description

Implements the Top Trading Cycle and Chains algorithm proposed by Roth et al. (2004) for the kidney exchange problem. The algorithm requires a rule to determine which chain will be used if there is more than one possibility. The chosen rule is to search for the longest chain and remove it from the problem (even the first kidney which was unassigned).

Usage

ttcc(nPatients = ncol(prefs), prefs, priority = NULL, seed = NULL)
ttcc(nPatients = ncol(prefs), prefs, priority = NULL, seed = NULL)

Arguments

`nPatients`	integer indicating the number of patient/donor-pairs in the matching problem. Defaults to `ncol(prefs)`.
`prefs`	matrix of dimension (`nPatients` + 1) x `nPatients` with column j containg patients jth ranking over kidneys in decreasing order of preferences (i.e. most preferred first). An entry with value (`nPatients` +1) indicates that the patient prefers the waiting list to all kidney below in his ranking (therefore they do not matter and can be neglected/NA).
`priority`	(Optional) vector of length `nStudents`. Gives the prioirity ordering of the students in the search for cycles (Do not confuse it with the preferences!), if nothing is specified a random ordering is chosen.
`seed`	(Optional) integer setting the state for random number generation. Defaults to seed = NULL

Value

ttcc returns a list with the matching and a vector containing the patients who are assigned to the waiting list. The matching comprises a data frame of the operations to be performed between patient-donor pairs (ind-obj).

Author(s)

Thilo Klein, Alexander Sauer

References

Roth, A.; T. Sonmez; U. Unver (2004). Kidney Exchange. Quarterly Journal of Economics, 119 (2): 457-488.

Examples

##\dontrun{
## Compare Example 1 from Roth et al. (2004) on page 469 - 475
## generate matrix of patients' preference rankings over kidneys, a.k.a. Rank Order Lists (ROL)

prefs <- matrix(c( 9,10, 1,NA,NA,NA,NA,
                  11, 3, 5, 6, 2,NA,NA,
                   2, 4, 5, 6, 7, 8,13,
                   5, 9, 1, 8,10, 3,13,
                   3, 7,11, 4, 5,NA,NA,
                   3, 5, 8, 6,NA,NA,NA,
                   6, 1, 3, 9,10, 1,13,
                   6, 4,11, 2, 3, 8,NA,
                   3,11,13,NA,NA,NA,NA,
                  11, 1, 4, 5, 6, 7,13,
                   3, 6, 5,11,NA,NA,NA,
                  11, 3, 9, 8,10,12,NA),
              byrow = FALSE, ncol = 12); prefs
priority <- 1:12
ttcc(prefs = prefs, priority = priority)
## The final matching differs slightly because in Round 3 another chain is chosen due to a different
## decision rule (compare Figure 3, p472. Here W1 instead of W2 is chosen)
##}
##\dontrun{
## Compare Example 1 from Roth et al. (2004) on page 469 - 475
## generate matrix of patients' preference rankings over kidneys, a.k.a. Rank Order Lists (ROL)

prefs <- matrix(c( 9,10, 1,NA,NA,NA,NA,
                  11, 3, 5, 6, 2,NA,NA,
                   2, 4, 5, 6, 7, 8,13,
                   5, 9, 1, 8,10, 3,13,
                   3, 7,11, 4, 5,NA,NA,
                   3, 5, 8, 6,NA,NA,NA,
                   6, 1, 3, 9,10, 1,13,
                   6, 4,11, 2, 3, 8,NA,
                   3,11,13,NA,NA,NA,NA,
                  11, 1, 4, 5, 6, 7,13,
                   3, 6, 5,11,NA,NA,NA,
                  11, 3, 9, 8,10,12,NA),
              byrow = FALSE, ncol = 12); prefs
priority <- 1:12
ttcc(prefs = prefs, priority = priority)
## The final matching differs slightly because in Round 3 another chain is chosen due to a different
## decision rule (compare Figure 3, p472. Here W1 instead of W2 is chosen)
##}

data.frame for exploded logit model

Description

shape a data.frame in a suitable form for the use of the exploded logit with the mlogit function in package mlogit.

Usage

xlogit.data(
  data,
  choice,
  shape = c("wide", "long"),
  varying = NULL,
  sep = ".",
  alt.var = NULL,
  chid.var = NULL,
  alt.levels = NULL,
  id.var = NULL,
  opposite = NULL,
  drop.index = FALSE,
  ranked = FALSE,
  ...
)
xlogit.data(
  data,
  choice,
  shape = c("wide", "long"),
  varying = NULL,
  sep = ".",
  alt.var = NULL,
  chid.var = NULL,
  alt.levels = NULL,
  id.var = NULL,
  opposite = NULL,
  drop.index = FALSE,
  ranked = FALSE,
  ...
)

Arguments

`data`	a data.frame
`choice`	the variable indicating the choice made: it can be either a logical vector, a numerical vector with 0 where the alternative is not chosen, a factor with level 'yes' when the alternative is chosen
`shape`	the shape of the data.frame: whether long if each row is an alternative or wide if each row is an observation
`varying`	the indexes of the variables that are alternative specific
`sep`	the seperator of the variable name and the alternative name (only relevant for a wide data.frame)
`alt.var`	the name of the variable that contains the alternative index (for a long data.frame only) or the name under which the alternative index will be stored (the default name is alt)
`chid.var`	the name of the variable that contains the choice index or the name under which the choice index will be stored
`alt.levels`	the name of the alternatives: if null, for a wide data.frame, they are guessed from the variable names and the choice variable (both should be the same), for a long data.frame, they are guessed from the alt.var argument
`id.var`	the name of the variable that contains the individual index if any
`opposite`	returns the opposite of the specified variables
`drop.index`	should the index variables be dropped from the data.frame
`ranked`	a logical value which is true if the response is a rank
`...`	further arguments passed to reshape

Value

xlogit.data returns a mlogit.data object, which is a data.frame in long format, i.e. one line for each alternative. It has a index attribute, which is a data.frame that contains the index of the choice made ('chid'), the index of the alternative ('alt') and, if any, the index of the individual ('id'). The choice variable is a boolean which indicates the choice made. This function use reshape if the data.frame is in wide format.

Author(s)

Thilo Klein

Examples

## Not run: 
## Game2 is a data.frame in long format for which the response is a
## ranking variable (see example in package mlogit, function mlogit.data)

## --- 1. baseline case (equivalent to mlogit package)

## load data
data("Game2", package = "mlogit")
head(Game2, 13)

## transform data
G <- xlogit.data(Game2, shape = "long", choice = "ch", 
                 alt.var = "platform", ranked = TRUE)
head(G, 30)

## --- 2. handle missing values 

## introduce NAs
Game2$ch[Game2$ch != 1] <- NA

## transform data
G <- xlogit.data(Game2, shape = "long", choice = "ch", alt.var = "platform", ranked = TRUE)
head(G, 30)

## --- 3. handle varying choice sets 

## drop choice options
Game2 <- Game2[-c(7:8),]

## transform data
G <- xlogit.data(Game2, shape = "long", choice = "ch", alt.var = "platform", ranked = TRUE)
head(G, 30)

## --- 4. run models
summary(mlogit::mlogit(ch ~ own | -1 + hours + age, G, reflevel = "PC"))
summary(mlogit::mlogit(ch ~ 0 | -1 + age | own, G))
summary(mlogit::mlogit(ch ~ 0 | -1 + age, G))

## End(Not run)
## Not run: 
## Game2 is a data.frame in long format for which the response is a
## ranking variable (see example in package mlogit, function mlogit.data)

## --- 1. baseline case (equivalent to mlogit package)

## load data
data("Game2", package = "mlogit")
head(Game2, 13)

## transform data
G <- xlogit.data(Game2, shape = "long", choice = "ch", 
                 alt.var = "platform", ranked = TRUE)
head(G, 30)

## --- 2. handle missing values 

## introduce NAs
Game2$ch[Game2$ch != 1] <- NA

## transform data
G <- xlogit.data(Game2, shape = "long", choice = "ch", alt.var = "platform", ranked = TRUE)
head(G, 30)

## --- 3. handle varying choice sets 

## drop choice options
Game2 <- Game2[-c(7:8),]

## transform data
G <- xlogit.data(Game2, shape = "long", choice = "ch", alt.var = "platform", ranked = TRUE)
head(G, 30)

## --- 4. run models
summary(mlogit::mlogit(ch ~ own | -1 + hours + age, G, reflevel = "PC"))
summary(mlogit::mlogit(ch ~ 0 | -1 + age | own, G))
summary(mlogit::mlogit(ch ~ 0 | -1 + age, G))

## End(Not run)

Package 'matchingMarkets'

Help Index

An R package for the analysis of stable matchings.

Description

Frequently Asked Questions

How to cite this package

Author(s)

References

See Also

Townsend Thai Project BAAC Annual Resurvey, 2000

Description

Usage

Format

Source

References

All stable matchings in the hospital/residents problem with incomplete lists

Description

Usage

Arguments

Value

Minimum required arguments

Author(s)

References

Examples

Resident-optimal matching in the hospital/residents problem with couples

Description

Usage

Arguments

Value

Minimum required arguments

Author(s)

References

Examples

Immediate Acceptance Algorithm (a.k.a. Boston mechanism) for two-sided matching markets

Description

Usage

Arguments

Value

Minimum required arguments

Author(s)

References

Examples

Karlson-Holm-Breen method for comparing probit coefficients

Description

Usage

Arguments

Author(s)

References

Examples

MCMC results in Klein (2015a)

Description

Usage

Format

References

Results of Monte Carlo Simulations in Klein (2015b)

Description

Usage

Format

References

Examples

MC Experiments

Description

Usage

Arguments

Author(s)

Examples

Partitioning Linear Programme for the stable roommates problem

Description

Usage

Arguments

Value

Author(s)

References

Examples

Predict method for fitted matching models

Description

Usage

Arguments

Author(s)

References