The assignment game: Decentralized dynamics, rate of convergence, and equitable core selection

Transcription

1 1 / 29 The assignment game: Decentralized dynamics, rate of convergence, and equitable core selection Bary S. R. Pradelski (with Heinrich H. Nax) ETH Zurich October 19, 2015

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3 3 / 29 Two-sided, one-to-one matching Non-transferable utility Pairwise stable outcomes always exist (Gale & Shapley 1962) Transferable utility Pairwise stable and optimal outcomes (core) always exist (Shapley & Shubik 1972) f f 1 2 ( w1 > w2 > w3) ( w3 > w2 > w1) f f 1 2 (76,62,40) (40,62,80) ( f > f) ( f1 f2) ( f1 > f2) 2 1 w1 w2 w3 (40,60) (40,40) (60,40) w1 w2 w3

4 Centralized matching markets How do matching markets equilibrate? Algorithm by central planner yields stable and optimal outcomes, e.g., deferred acceptance algorithm (Gale & Shapley 1962, Crawford & Knoer 1981) solving LP-dual (Shapley & Shubik 1972) Planner knows agents and their preferences / reservation values Use in practice Resource allocations for hospitals, transplantations or school admissions (Roth & Sotomayor 1990, Roth et al....) 4 / 29

5 5 / 29 Decentralized matching markets New internet and communication technologies create large decentralized markets, e.g., Online markets for matching buyers and sellers of goods (ebay), labor markets (mechanical turk) Market decentralization agents interact, match, and break up at random points in time Information decentralization agents have no information about preferences / reservation values of other agents

6 6 / 29 How do decentralized matching markets equilibrate? Dynamic. Pairs randomly meet and match if they are both better off Transferable utility 1 Convergence to stable and optimal outcomes? Yes 2 Rate of convergence? Polynomial if correlated shocks 3 Robustness / selection? Equitable selection We show this by studying a simple aspiration-based learning rule.

7 7 / 29 Aspiration-based learning Learning Boundedly rational agents update their behavior in the light of experience with little information: completely uncoupled (Hart & Mas-Colell 2003, Foster & Young 2006) Aspiration-based learning A subject lowers his aspiration level after a negative impulse, keeps his aspiration level after a neutral impulse, and raises his aspiration level after a positive impulse (Hoppe 1931, Sauermann & Selten 1962, tested by Tietz, Weber, et al. in the 70 s and 80 s)

8 8 / 29 Dynamic model: every period an agent is activated Matched agent i Single agent i meets profitable match meets no profitable match meets profitable match meets no profitable match new match new aspiration level old match old aspiration level new match new aspiration level no match reduced aspiration level

9 9 / 29 Firms/workers and their willingness to pay/accept Firms i F and workers j W repeatedly look for partners ( F = W = N) r ( j) + i δn Firm i is willing to pay at most r + i (j) δn to match worker j Worker j is willing to accept at least r j (i) δn to match firm i where δ> 0 is the minimum unit ( dollars ) The willingness to pay/accept is private information, it is not known to other agents. r () i j 0

10 10 / 29 The match value δn The match value for the pair (i, j) is r ( j) + i α ij = (r + i (j) r j (i)) + Let α = (α ij ) i F,j W The match value is a hidden variable. α ij r () i j 0

11 11 / 29 Aspiration levels together with an assignment define the state Let t = 0, 1, 2,... be the time periods. At the end of period t each agent has an aspiration level d t i 0, let dt = (d t i ) i F W r ( j) + i δn t d i Let A t = (a t ij ) i F,j W be the assignment such that each agent has at most one partner at a time and { matched then a t ij if (i, j) is = 1 unmatched then a t ij = 0 α ij t d j At the end of period t the state is given by r () i j Z t = [A t, d t ] 0

12 12 / 29 Activation and bidding In period t + 1, a random agent is activated, say i F, and makes a random encounter, say j W. Each makes a bid to match with the other, r ( j) + i δn t d i t+1 p ij i offers p t+1 ij and j asks q t+1 ij such that an agent s bid, if realized, yields at least as much as his aspiration level: p t+1 ij q t+1 ij r + i (j) d t i r j (i) + d t j α ij r () i j t d j t+1 q ij and with positive probability equality holds. 0

13 13 / 29 Profitability, price, and payoff Two bids are profitable if each agent, in expectation, receives a higher payoff from the match than his previous-period payoff. r ( j) + i δn Only profitable pairs match. When i matches with j the price is set at random on δn with full support such that q t+1 ij π t+1 ij p t+1 ij α ij t+1 p ij t+1 q ij π t+1 ij r () i j 0

14 13 / 29 Profitability, price, and payoff Two bids are profitable if each agent, in expectation, receives a higher payoff from the match than his previous-period payoff. r ( j) + i δn Only profitable pairs match. When i matches with j the price is set at random on δn with full support such that q t+1 ij π t+1 ij p t+1 ij α ij π t+1 ij φ t+1 i φ +1 t j The payoff to firm i is φ t+1 i = r i + (j) πij t+1 The payoff to worker j is φ t+1 j = πij t+1 rj (i) If an agent i is single φ t+1 i = 0. r () i j 0

15 14 / 29 New assignment and new aspiration levels If (i, j) newly matched set a t+1 ij = 1 and their previous partners become single; i and j update their aspiration levels r ( j) + i δn di t+1 = φ t+1 i, dj t+1 = φ t+1 j t +1 d i φ t+1 i If i does not rematch and if he was matched in t, Z t+1 = Z t. If he was single in t he updates his aspiration level where X t+1 i d t+1 i = (d t i X t+1 i ) + δn is a RV. α ij r () i j t +1 d j φ +1 t j The new state is Z t+1 = [A t+1, d t+1 ]. 0

16 15 / 29 Solution concepts Optimality. A is optimal if it maximizes total payoff: (i,j) F W a ij α ij Pairwise stability. d is pairwise stable if for all (i, j) matched and for all k, l d i + d j = α ij d k l a kl + d l k a kl α kl Core. The core of an assignment game consists of all states, [A, d], such that A is an optimal assignment and d is pairwise stable. The core of the assignment game is nonempty (Shapley & Shubik 1972).

17 16 / 29 Theorem 1 (Nax & Pradelski 2014) Given an assignment game, from any initial state [A 0, d 0 ], the process converges to the core in finite time with probability 1.

18 17 / 29 Example Lines indicate possible matches. The vector next to an agent represents his willingness to pay/accept. Solid edges indicate current matches; dashed potentially profitable matches. Next to an agent is his aspiration level; next to an edge is the match value α ij (if positive). f f 1 2 (76,62,40) (40,62,80) f f 1 2 d f1 d f (40,60) (40,40) (60,40) w1 w2 w3 d w1 d w2 d w3 w1 w2 w3

19 18 / 29 Period-t state: Z t suppose the transfers are discretized with minimum unit 1 f f w1 w2 w3

20 19 / 29 Period t + 1 activation of single agent w 3 f f w1 w2 w3

21 19 / 29 Period t + 1 w 3 encounters f 2 f f w1 w2 w3

22 19 / 29 Period t + 1 the two agents make bids for each other since p t and q t the bids are not profitable f f = = 59 w1 w2 w3

23 19 / 29 Period t + 1 w 3 remains single and, with positive probability, reduces his aspiration level by 2 f f w1 w2 w3

24 19 / 29 Period t + 1 at the end of the period Z t+1 is f f w1 w2 w3

25 20 / 29 Period t + 2 activation of matched agent f 2 f f w1 w2 w3

26 20 / 29 Period t + 2 f 2 encounters w 3 f f w1 w2 w3

27 20 / 29 Period t + 2 the two agents make bids for each other with positive probability p t+1 23 = 58 and q t+1 23 = 57; the bids are profitable f f = = 57 w1 w2 w3

28 20 / 29 Period t + 2 with positive probability the price is set to 57, the agents match and update their aspiration levels f f = = 17 w1 w2 w3

29 20 / 29 Period t + 2 at the end of the period Z t+2 is in the core f f w1 w2 w3

30 Core geometrical representation It suffices to consider the aspiration level space of one side of the market, spanned by the equations d f1 + d w1 = 36 and d f2 + d w3 = d f2 firm optimal core 22 worker optimal d f1 21 / 29

31 22 / 29 Market condition and price stickiness Let M t {, } be a binary random variable describing the market condition at time t: If M t =, the firms exhibit price stickiness, e.g., in periods of high unemployment If M t =, the workers exhibit price stickiness, e.g., in periods of low unemployment

32 22 / 29 Market condition and price stickiness Let M t {, } be a binary random variable describing the market condition at time t: If M t =, the firms exhibit price stickiness, e.g., in periods of high unemployment If M t =, the workers exhibit price stickiness, e.g., in periods of low unemployment How price stickiness influences the dynamics an active agent enters negotiations with a single if he is equally well-off given the bids a single matches below his bid if he is currently not price sticky and has no equally good alternative

33 23 / 29 Theorem 2 (Pradelski 2015) Given an assignment game with discrete generic weight matrix: Suppose that M t switches every Θ(N 2+k ), (k 0) time steps: The expected rate of convergence to the core is O(N 4+k ). Else convergence to the core is 2 Θ(N). A match value matrix α is discrete generic if the corresponding graph has no cycles such that two alternating partitions have the same sum of weights.

34 24 / 29 Random perturbations Matched agents experience iid payoff shocks. For matched agent i the perturbed payoff in t is { ˆφ t i = φ t i + δ Rt i with probability 0.5 φ t i δ Rt i with probability 0.5 R t i is a geometric random variable: P[R t i = k] = ε k (1 ε) for all k N 0 Thus an agent may get a different payoff than anticipated given the current price.

35 25 / 29 Solution concepts Excess. Given state Z t, the excess for an agent i matched with j is e t i = φt i max k j(α ik φ t k ) + The minimal excess across all matched agents is e t min (Zt ) = min i e t i Least core. (Maschler et al. 1979) The least core of an assignment game is the set of states such that the matching is optimal and the minimal excess is maximized. For assignment games the least core is contained in the core.

36 26 / 29 Theorem 3 (Nax & Pradelski 2014) Given an assignment game, the stochastically stable states are contained in the least core. The least core consists of the states which are most robust to single payoff shocks.

37 27 / 29 Least core geometrical representation The least core is such that d f1 = 29 and d f2 = 29,..., d f2 firm optimal least core 22 worker optimal d f1

38 28 / 29 Conclusion We study a two-sided market for heterogeneous goods/buyers. Despite the severe informational restrictions in decentralized markets: 1 Convergence to the core? Yes 2 Rate of convergence? Polynomial if correlated shocks 3 Robustness / selection? Equitable selection

39 28 / 29 Conclusion We study a two-sided market for heterogeneous goods/buyers. Despite the severe informational restrictions in decentralized markets: 1 Convergence to the core? Yes 2 Rate of convergence? Polynomial if correlated shocks 3 Robustness / selection? Equitable selection THANK YOU!

40 29 / 29 Selected citations Non-transferable utility D. Gale & L. S. Shapley (1962), College admissions and the stability of marriage, American Mathematical Monthly, 69, 9-15 A. E. Roth & H. Vande Vate (1990), Random paths to stability in two-sided matching, Econometrica, 58, H. Ackermann et al. (2011), Uncoordinated two-sided matching markets, SIAM Journal on Computing, 40, Transferable utility L. S. Shapley & M. Shubik (1972), The Assignment Game I: The Core, International Journal of Game Theory 1, M. Maschler et al. (1979), Geometric properties of the kernel, nucleolus, and related solution concepts, Mathematics of Operations Research, 4, Learning S. Hart & A. Mas-Colell (2003), Uncoupled dynamics do not lead to Nash equilibrium, American Economic Review, 93, D. Foster & H. P. Young (2006), Regret testing: Learning to play Nash equilibrium without knowing you have an opponent, Theoretical Economics 1, This talk H. H. Nax & B. S. R. Pradelski (2014), Evolutionary dynamics and equitable core selection in assignment games, International Journal of Game Theory, available online B. S. R. Pradelski (2015), Decentralized dynamics and fast convergence in the assignment game, extended abstract in EC 2015