Stable Marriage and Search Frictions ¨ Stephan Lauermann and Georg Noldeke ¨ Basel University of Michigan and Universitat
July 5, 2012
1. Introduction
We consider a marriage market: two-sided matching market with nontransferable utility. General preferences as in Gale and Shapley (1962). Aim: Use search model as a test-bed to investigate intuition that stable matchings are the appropriate solution concept when frictions are negligible. Main Result: Answer depends on whether the marriage market has a unique stable matching or not. Uniqueness implies that all equilibrium matchings in the search model approximate the stable matching when frictions are small. If there are multiple stable matchings, the set of limit matchings contains the set of all stable deterministic matchings, but also contains unstable random matchings. ¨ Lauermann and Noldeke,
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2. Related Literature Eeckhout (1999) and Adachi (2003) also consider relationship between stable matchings in a marriage market and equilibrium matchings in a search model when frictions become negligible. The underlying marriage market in Eeckhout has a unique stable matching. Adachi does not consider the possibility of random limit matchings. Both consider convergence in the context of a “cloning” model.
Marriage Market with Search Frictions: McNamara and Collins (1990), Burdett and Coles (1997), Bloch and Ryder (2000), Smith (2006), . . . . Literature on convergence to competitive equilibria in dynamic matching and bargaining games (Rubinstein and Wolinsky 1985, Gale 1987 . . . ) studies, in effect, a special case of the TU version of our problem. (Lauermann 2012) ¨ Lauermann and Noldeke,
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3. Marriage Market
Two disjoint, finite set of agents: M and W . Agents have strict preferences over the agents in the other set and the prospect of remaining single. u(m, w): utility of agent m ∈ M from being matched with w ∈ W . v(m, w): utility of agent w ∈ W from begin matched with m ∈ M. Utility from being unmatched is normalized to zero. Assume there is at least one pair of agents such that u(m, w) > 0 and v(m, w) > 0 holds.
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3. Marriage Market A matching is described by an assignment matrix x with elements x(m, w) satisfying
∑ x(m, w) ≤ 1 for all m ∈ M,
w∈W
∑ x(m, w) ≤ 1 for all w ∈ W,
m∈M
x(m, w) ≥ 0 for all (m, w) ∈ M ×W. Think of x(m, w) as specifying a probability that m and w are matched. A matching is deterministic if x(m, w) ∈ {0, 1} for all (m, w); otherwise it is random. Every random matching can be generated by a lottery over the set of deterministic matchings (Birkhoff-von Neumann theorem). ¨ Lauermann and Noldeke,
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3. Marriage Market
Payoffs associated with a matching given by: U(m; x) =
∑ x(m, w)u(m, w)
for all m ∈ M,
∑ x(m, w)v(m, w)
for all w ∈ W.
w∈W
V (w; x) =
m∈M
Define x(m, m) = 1 −
∑ x(m, w) and x(w, w) = 1 − ∑ x(m, w).
w∈W
m∈M
x(h, h) = 0: h is fully matched. x(h, h) = 1: h is unmatched. 0 < x(h, h) < 1: h is partially matched. ¨ Lauermann and Noldeke,
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3. Marriage Market
A matching is individual rational if x(m, w) > 0 ⇒ u(m, w) ≥ 0 and v(m, w) ≥ 0. A matching is pairwise stable if there does not exist (m, w) ∈ M ×W satisfying u(m, w) > U(m; x) and v(m, w) > V (w; x). We say that a matching is coherent if it is individual rational and pairwise stable.
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3. Marriage Market
A deterministic matching is coherent if and only if it is stable in the sense of Gale and Shapley (1962). A random matching is stable (Vande Vate 1989) if and only if it is a convex combination of stable deterministic matchings (Roth, Rothblum, Vande Vate 1993). For random matchings coherency is neither necessary nor sufficient for stability. In a stable random matching all agents are either fully matched or unmatched, whereas coherent matchings may have partially matched agents. Example 1 Not every stable random matching satisfies pairwise stability. Example 2
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3. Marriage Market A matching is regret-free if x(m, w) > 0 ⇒ u(m, w) ≥ U(m; x) and v(m, w) ≥ V (w; x).
Lemma 1. Stable deterministic matchings are regret-free. 2. Stable random matchings are not regret-free. Proof: 1. Every deterministic matching is regret free. 2. In a stable random matching all agents are either fully matched or unmatched. In a regret-free random matching there are partially matched agents. ¨ Lauermann and Noldeke,
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4. Search
Continuous time random search model. Mass η > 0 of agents of each type m ∈ M resp. w ∈ W are “born” and start searching per unit time. For the purpose of this talk set η = 1.
Upon meeting a potential partner agents decide whether or not to form a match. No match: Agents continue searching. Match: Agents stop searching and receive the payoffs u(m, w) and v(m, w).
At rate δ > 0 agents “die” and stop searching. Such agents are unmatched and receive payoff zero.
No (further) discounting.
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4. Search
For the purpose of this talk, assume that meetings are generated by a quadratic search technology: f (m) > 0: mass of agents of type m searching for a partner. g(w) > 0: mass of agents of type w searching for a partner. The mass of pairs (m, w) that meet per unit time is λ f (m)g(w) > 0. λ > 0 is an exogenous parameter, measuring the speed of the meeting process.
Let 0 ≤ α(m, w) ≤ 1 denote the proportion of meetings between agents of type m and w that result in a match.
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4. Search
Definition (Steady State) ( f , g, α) is a steady state if " 1 = f (m) δ + λ
#
∑ α(m, w)g(w)
for all m ∈ M,
w∈W
" 1 = g(w) δ + λ
#
∑ α(m, w) f (m)
for all w ∈ W.
m∈M
The left side of these equations represents the inflow of newborn agents of a given type. The right hand side is the corresponding outflow of matched and single agents. ¨ Lauermann and Noldeke,
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4. Search
Definition (Steady State Matching) A matching x is a steady state matching if it is induced by some steady state ( f , g, α), that is, x(m, w) = λ α(m, w) f (m)g(w) for all (m, w) ∈ M ×W.
Steady state conditions ensure that x(m, w) as defined above is the probability that a newborn of type m (resp. w) exits the market in a match with an agent of type w (resp m). Corresponding steady state payoffs are given by U(m; x) and V (w; x).
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4. Search
Definition (Equilibrium) A steady state ( f , g, α) is an equilibrium if ( 0 if u(m, w) < U(m; x) or v(m, w) < V (w; x), α(m, w) = 1 if u(m, w) > U(m; x) and v(m, w) > V (w; x) holds for all (m, w) ∈ M ×W , where x is the matching induced by ( f , g, α). Any α(m, w) ∈ [0, 1] is consistent with equilibrium if u(m, w) ≥ U(m; x) and v(m, w) ≥ V (w; x) are both satisfied and (at least) one of them holds with equality.
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4. Search
Definition (Equilibrium Matching) A matching x is an equilibrium matching if it is induced by some equilibrium ( f , g, α).
Lemma Every equilibrium matching is individual rational and regret-free. Equilibrium matchings exist. Due to the possibility of exit as a single, all equilibrium matchings are random and unstable.
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5. Limit Matchings
Definition (Limit Matching) A matching x∗ is a limit matching if there exists a sequence (λk , xk ) satisfying λk → ∞. xk → x∗ . xk is an equilibrium matching for the search model with parameter λk .
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5. Limit Matchings
Proposition (Characterization of Limit Matchings) x∗ is a limit matching if and only if it is coherent and regret-free.
Corollary (Stability of Limit Matchings) 1. A deterministic matching is a limit matching if and only if it is stable. 2. Every random limit matching is unstable. Proof of Corollary: Immediate from the stability properties of regret-free matchings.
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5. Limit Matchings
Main issues in proving the Proposition: 1. Because every equilibrium matching is individual rational and regret-free, it is immediate that every limit matching has these properties. Pairwise stability of limit matchings is intuitive, but less obvious. Why isn’t this obvious? 2. The proof that every stable deterministic matching is a limit matching is intuitive and easy. Proving that every coherent and regret-free random matching is a limit matching is harder. Example 1
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5. Limit Matchings
Random limit matchings are unstable because there exists a pair of agents (m, w) such that both m and w are partially matched, even though m and w find each other acceptable. For the same reason random limit matchings are also Pareto-dominated. Remaining Question: Do random limit matchings exist?
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5. Limit Matchings
From previous results: The set of random limit matchings is identical to the set of random matchings x that are 1. individual rational: x(m, w) > 0 ⇒ u(m, w) ≥ 0 and v(m, w) ≥ 0. 2. regret-free: x(m, w) > 0 ⇒ u(m, w) ≥ U(m; x) and v(m, w) ≥ V (w; x). 3. pairwise stable: there does not exist (m, w) satisfying u(m, w) > U(m; x) and v(m, w) > V (w; x).
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5. Limit Matchings
Proposition (Existence of Random Limit Matchings) Random limit matchings exist if and only if there are multiple stable deterministic matchings. Key steps in the proof: Starting from any coherent and regret-free random matching, two distinct stable deterministic matchings can be constructed by assigning all men (resp. all women) to their best partner among those satisfying x(m, w) > 0. Vice versa, given any two consecutive stable deterministic matchings, a random matching with the requisite properties can be constructed in which x(m, w) > 0 holds if and only if (m, w) are matched in one of the stable deterministic matchings. Example 2 ¨ Lauermann and Noldeke,
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5. Limit Matchings
Corollary (Stability of All Limit Matchings) All limit matchings are stable if and only if there is a unique stable matching. Two prominent examples with unique stable matchings: Vertical Heterogeneity: The sets M and W are ordered such that all agents prefer partners of higher rank. Horizontal Heterogeneity: Each individual is the ideal partner of his/her own ideal partner.
More general conditions for uniqueness are given in Eeckhout (Economics Letters 2000) and Clark (Contributions to Theoretical Economics 2006).
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6. Summary
Complete characterization of limit matchings in the search model. Stability obtains for deterministic limit matchings, but fails for random limit matchings. Random limit matchings exist unless there is a unique stable matching.
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7. Topics for Discussion
Importance of “technical” assumptions such as quadratic meeting technology, absence of explicit discounting, strictness of preferences, and absence of a tiebreaking rule. Extension to the roommate problem. Time sharing and rematching. Extension to “partially transferable” utility (Legros and Newman 2007).
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Example 1 M = {m1 , m2 } and W = {w1 , w2 }. Utilities given by the bi-matrix w1 m1 2, 1 m2 1, 2
w2 1, 2 2, 1
Two stable deterministic matchings given by: 0 1 1 0 x1 = and x2 = 0 1 1 0 Stable random matchings given by α 1−α 1−α α with 0 < α < 1. ¨ Lauermann and Noldeke,
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Example 1
In this example all stable matchings are coherent, but there are additional coherent matchings. For instance, the random matching 1/3 1/3 x∗ = 1/3 1/3 with payoffs U(m1 ; x∗ ) = U(m2 ; x∗ ) = V (w1 ; x∗ ) = V (w2 ; x∗ ) = 1. is coherent, but not stable.
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Example 1
The random matching x∗ from the previous slide is coherent and regret-free. To prove that x∗ is a limit matching, it suffices to show that for all λ high enough it is an equilibrium matching. Think of agents as following the strategy to accept his or her favorite partner whenever they meet and accepting the less attractive partner with a probability 0 < a < 1. For λ high enough, there exists a value of a such that in the corresponding steady state all agents are indifferent between accepting or rejecting the less attractive partner. This steady state is an equilibrium and induces x∗ .
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Example 2
M = {m1 , m2 , m3 } and W = {w1 , w2 , w3 }. Utilities given by the bi-matrix m1 m2 m3
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w1 7, 3 3, 7 6, 6
w2 6, 6 7, 3 3, 7
w3 3, 7 6, 6 7, 3
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Example 2
There are three stable deterministic matchings given by 1 0 0 0 1 0 0 0 1 x1 = 0 1 0 , x2 = 0 0 1 , and x3 = 1 0 0 0 0 1 1 0 0 0 1 0
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Example 2
The random matching xˆ = 0.5x1 + 0.5x2 is stable, but not coherent: u(m1 , w2 ) = 6 > U(m1 ; x) ˆ = 5 and v(m1 , w2 ) = 6 > V (w2 ; x) ˆ = 5.
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Example 2
Besides the stable deterministic matchings there are exactly two fractional limit matchings, namely x∗ =
18 3 18 3 x1 + x2 and x∗∗ = x3 + x2 . 24 24 24 24
In x∗ payoffs are U(mi ; x∗ ) = 6 and V (wi ; x∗ ) = 3. In x∗∗ payoffs are U(mi , x∗∗ ) = 3 and V (wi ; x∗∗ ) = 6.
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Pairwise Stability of Limit Matchings
If the pairwise stability condition is violated in a limit matching x∗ , then there exists a pair of agents (m0 , w0 ) such that u(m0 , w0 ) > U(m0 ; x∗ ) and v(m0 , w0 ) > V (w0 , x∗ ). Provided that λk fk (m0 ) or λk gk (w0 ) goes to infinity, this contradicts the equilibrium conditions for k large enough. The difficulty is in showing that at least one of the above contact rates converges to infinity In a cloning model this is trivial because masses are exogenous. In our model fk (m0 ) and gk (w0 ) will both converges to zero . . ..
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