1 Sequential Correlated Equilibria in Stopping Games Yuval Heller Tel-Aviv University (Part of my Ph.D. thesis supervised by Eilon Solan) Stony-Brook July 2010

Contents  Introduction Motivating example Solution concept Simplifying assumptions  Model: stopping games  Main result: equilibrium existence Proof’s sketch 2

Motivating Example  Monthly news release on U.S. employment situation published in the middle of the European trading day Strong impact on the stock markets (Nikkinen et al., 06) Market’s adjustment lasts tens of minutes (Christie-David et al., 02)  Strategic interaction between traders of an institution Common objective: maximize the institution's profit Private objective: maximize “his” profit (bonuses, prestige)  Traders can freely communicate before the announcement, but later communication is costly 3

Strategic Interaction - Properties  Interaction lasts short absolute time, but agents have many instances to act. Ending time may be unknown in real-time Infinite-horizon (noncooperative) game, undiscounted payoffs (Rubinstein, 91; Aumann & Maschler, 95)  Agents share similar, though not identical goals  Agents may occasionally make mistakes  Agents can freely communicate pre-play, but communication along the play is costly 4

Solution Concept (1)  Uniform equilibrium An approximate equilibrium in any long enough finite-horizon game (Aumann & Maschler, 95)  Sequential equilibrium (Kreps & Wilson, 82) Players may make mistakes Behavior should be rational also after a mistake  Normal-form correlation (Forges, 1986): Pre-play communication is used to correlate players’ strategies before the play starts (Ben-Porath, 98) 5

Solution Concept (2)  Player’s expected payoff doesn’t depend on communication Facilitate implementation of pre-play coordination Constant-expectation correlated equilibrium Generalizes distribution equilibrium (Sorin, 1998)  Approximate ( ,  ) equilibrium: With probability at least 1- , no player can earn more than  by deviating after any history  Our solution concept: sequential uniform constant- expectation normal-form correlated ( ,  )-equilibrium 6

Simplifying Assumptions  Players have symmetric information Each trader can electronically access data on prices in all markets  Each player has a finite number of actions Each trader has a finite set of financial instruments, and for each he chooses time to buy & time to sell 7

Examples for Applications (1)  Several countries ally in a war Allying countries share similar, but not identical goals War lasts a few weeks but consists large unknown number of stages Leaders can coordinate strategies in advance. Secure communication during the war may be costly/noisy A few battlefield actions are crucial to the war’s outcome 8

Examples for Applications (2)  Male animals compete over positions in pack order (Maynard Smith, 74) Ritualized fighting, no serious injuries Excessive persistence – waste of time & energy Winner – last contestant Lasts a few hours/days; large unknown number of stages Normal-form correlation can be induced by phenotypic conditional behavior (Shmida & Peleg, 97) Constant-expectation requirement is needed for population stability (Sorin, 98) 9

Model Stopping Games 10

Multi-Player Stopping Games (1)  Players receive symmetric partial information on an unknown state variable along the game General (not-necessarily finite) filtration  Each player i may take up to T i <  actions during the game  At stage 1 all players are active  At every stage n each active player simultaneously declares if he takes one of a finite number of actions or “does nothing” 11

Multi-Player Stopping Games (2)  Player who acted T i times become passive for the rest of the game and must “do nothing”  Payoff depends on the history of actions and on the state variable  Reductions (equilibrium existence): T i =1 for all players (by induction ) Each player has a single “stopping” action Game ends as soon as any player stops 12

Stopping Games: Related Literature (1)  Introduced by Dynkin (1969)  Applications: Research & development Fudenberg & Tirole (85), Mamer (87) Firms in a declining market (Fudenberg & Tirole, 86) Auctions (2 nd price all-pay, Krishna & Morgan, 97) Lobbying (Bulow & Klemperer, 01) Conflict among animals (Nalebuff & Riley, 85) T i >1 case: Szajowski (02), Yasuda & Szajowski (02), Laraki & solan (05) 13

Stopping Games: Related Literature (2)  Approximate Nash equilibrium existence in undiscounted 2-player games: Neveu (75), Mamer (87), Morimoto (86), Nowak & Szajowski (99), Rosenberg, Solan & Vieille (01), Neumann, Rmasey & szajowski (02), Shmaya & Solan (04)  Undiscounted multi-player stopping games were mostly modeled as cooperative games Ohtsubo (95, 96, 98), Assaf & Samiel-Cahn (98), Glickman (04), Ramsey & Cierpial (09)  Mashiah-Yaakovi (2008) – existence of approximate perfect equilibrium when simultaneous stops aren’t allowed 14

Stopping Games with Voting procedures  Each player votes at each stage whether he wishes to stop  Some monotonic rule (e.g., majority) determines which coalitions can stop  Studied in: Kurano, Yasuda & Nakagami (80, 82), Szajowski & Yasuda (97)  Simplifying assumption: payoff depends only on the stopping stage but not on the stopping coalition  Our model: Does not assume this simplifying assumption Can be adapted to include a voting procedure 15

Main Result: Equilibrium Existence  A multiplayer stopping game admits a sequential uniform constant-expectation normal-form correlated (  -equilibrium (   >0)  Two appealing properties: Canonical – each signal is equivalent to a strategy Correlation device doesn’t depend on the specific game parameters 16

Proof’s Sketch 17

18 Games on Finite Trees  Simplifying assumption: finite filtration  Equivalent to a special kind of an absorbing game: stochastic game with a single non-absorbing state If not stopped earlier, game restarts at the leafs

Games on Finite Trees (Solan & Vohra, 02)  Every such game admits : 1. Stationary equilibrium, or 2. Correlated distribution  over action-profiles in which a single player stops: Player is chosen according to  and being asked to stop Incentive-compatible (correlated  -equilibrium.) 19

Games on Finite Trees: Strengthening Solan-Vohra result 1.Stationary sequential equilibrium Perturbed game with positive continuity probability 2.Modifying the procedure of asking to stop Ask to stop with probability 1-   sequentiality Players can’t deduce being off-equilibrium path When a player receives his signal he can’t deduce who has been asked to stop  constant expectation  Adapting the methods of Shmaya & Solan (04) to deal also with infinite filtrations 20

Concatenating Equilibria 21

22 Ramsey Theorem (1930)  A finite set of colors  Each two integers (k,n) are colored by c(k,n)  There is an infinite sequence of integers with the same color: k 1 <k 2 <k 3 <… such that: c(k 1,k 2 ) =c(k i,k j ) for all i<j k1k1

23 Stochastic Variation of Ramsey Theorem (Shmaya & Solan, 04)  Coloring each finite tree  There is an infinite sequence of stopping times with the same color:  1 1-  Low probability 11

Finishing the Proof  Each finite tree is colored according to the equilibrium payoff and properties  Using Shmaya-Solan’s theorem we concatenate equilibria with the same color  We verify that the induced profile is a (  -equilibrium with all the required properties 24

Summary  Solution concept for undiscounted dynamic games: sequential uniform constant-expectation normal- form correlated (  -equilibrium  Main result: every multi-player stopping game admits this equilibrium  Such games approximate a large family of interesting strategic interactions 25

26 Questions & Comments? Yuval Heller