# 1 Perfect Correlated Equilibria in Stopping Games Yuval Heller Tel-Aviv University (Part of my Ph.D. thesis supervised by Eilon Solan)

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1 Perfect Correlated Equilibria in Stopping Games Yuval Heller Tel-Aviv University (Part of my Ph.D. thesis supervised by Eilon Solan) http://www.tau.ac.il/~helleryu/ 3 rd Israeli Game Theory Conference December 2008

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Summary 2 Proof Outline Reductions Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction

3 Stopping Games (Undiscounted, Multi-player, Discrete time)  Finite set of players: I  Unknown state variable:     (state space) Filtration: F =( F n  )  At each stage n the players receive a symmetric partial information about the state : F n (  )

4 Stopping Games (undiscounted, multi-player, discrete time)  Stage 1 - everyone is active  Stage n: All active players simultaneously declare whether they stop or continue A player that stops become passive for the rest of the game Player’s payoff depends on the history of players’ actions while he has been active and on the state variable

5 Literature: 2-player zero-sum Stopping Games  Dynkin (1969) – introduction, value where simultaneous stops are not allowed  Neveu (1975) – value when each player prefers the other to stop  Rosenberg, Solan & Vieille (2001) – use of randomized strategies, value with payoffs’ integrability

6 Literature: 2-player non-zerosum Stopping Games  Existence of approximate Nash equilibrium when the payoffs have a special structure: Morimoto (86), Mamer (87), Ohtsubo (87, 91), Nowak & Szajowski (99), Neumann, Ramsey & Szajowski (02)  Recently, Shmaya & Solan (04) proved existence assuming only integrability  Multi-player stopping games: no existence results

7 Stopping Games - Applications  Most applications in the literature: Payoffs: Specific assumptions, such as monotony Discount factor 2 players Multi-player variations are natural

8 Struggle of survival in a declining market  At each turn, each firm loses money  A firm can stay or exit the market for good Partial production is inefficient Market is more profitable with less firms  Which firms survive? What is the exit order?  Ghemawat & Nalebuff (1985)...  Steel market in 70’s and 80’s

9 Research & Development  Race for developing a patent  At each turn, continue spending money on research or leave the race  The first firm to complete the patent earns a lot Stochastic function of spent money  Fudenberg & Tirole (1985)…

10 War of attrition  Attrition wars among animals: Becoming the leader (alpha-male) Territory Maynard-Smith (1982), Nalebuff & Riley (1985)…  2 nd price auctions where all bidders pay Krishna & Morgan (1997)….  Political Sciences – lobbying Bulow & Klemperer (2001)

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Summary 11 Proof Outline Reductions Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction

12 Perfect Equilibrium  Nash equilibrium may be sustained by non-credible threats of punishment Punisher receives a low payoff  The stronger concept of perfect equilibrium (Selten, 1965, 1975) has been studied. Examples: Fine & Li (1989): uniqueness in discounted 2-player games with monotone payoffs Mashiah-Yaakovi (2008) – existence of (  )-perfect equilibrium when simultaneous stops aren’t allowed

13 Correlated Equilibrium  Aumann (1974): An equilibrium in an extended game with a correlation device Device D sends each player i a private signal m i  M i (M=  i M i ) before the game starts according to    (M) The extended game G(D)  Consistent with Bayesian decision making (Aumann, 87)  Other appealing properties: computability, linear equations, closed and convex set

14 Correlated Equilibrium in Sequential Games  Two main versions: Normal-form: signals are sent only before the game starts Extensive-form: signals are sent at each stage  Equilibrium: normal-form  extensive-form  Correlation among players is natural in many setups: Countries negotiate actions Firms choose strategies based on market’s history A manager coordinates the actions of his workers

15 Normal-Form Correlation (1)  Sometimes players may coordinate before play starts but coordination along the play is costly / impossible:  Example (1) - war of attrition in nature: Commonly modeled as stopping games Coordination before play starts is implemented by evolution of phenotype roles E.g.: Shmida & Peleg, 1997

16 Normal-Form Correlation (2)  Example (2) - News playing among day traders: Monthly employment report will be published at noon Several minutes elapse before market adjusts New information gradually arrives during that time Quick trading can be profitable See e.g., Christie-David, Chaudhry & Khan (2002) Traders of a firm can coordinate their actions in advance Coordination along the play is costly (time limit) Traders may have different payoffs

17 (  )-Perfect Correlated Equilibrium   – A bound for the probability of: An event E   Correlation device sends a signal in M’  M    >0 – A bound for the maximal profit a player can earn by deviating at any stage and after any history, conditioned on that    E and m  M’  Extending the definitions for finite games: Myerson (1986), Dhillon & Mertens (1996)

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Summary 18 Proof Outline Reductions Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction

19 Main Result  For every    >0, a multi-player stopping game admits a normal-form uniform perfect correlated (   )-equilibrium with a universal correlation device Uniform: An approximate equilibrium in any long enough finite game and in any discounted game with high enough discount factor Universal device – doesn’t depend on game payoffs Corollary: Uniform perfect correlated equilibrium payoff

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Proof Outline Reductions Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 20 Summary

1.Terminating games: game terminates at the first stop 2.Tree-like games (Shmaya & Solan, 03) : for every n, F n is finite A finite collection of matrix payoffs 3.Deep enough in the tree: with high probability any matrix payoff either: Repeats infinitely often Never occurs 21 Reductions

22 Reductions  Reductions require 2 properties from the equilibrium ( ,  -unrevealing - expected payoff of each player “almost” doesn’t change With probability of at-least 1- , changes by less than  Universal - The correlation device D(G, ,  ) depends only on |I| and    D(G, ,  )=D(|I|,  )

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 23 Summary Reductions

24 Games on Finite Trees  Equivalent to an absorbing game: A stochastic game with a single non-absorbing state. 2 special properties: Recursive game – Payoff in non-absorbing states is 0 Single non-absorbing action profile

25 Games on Finite Trees  An adaptation of a result of Solan & Vohra (2002):  A game on a finite tree has one of the following: 1.Non-absorbing equilibrium (game never stops) 2.Stationary absorbing equilibrium. Adaptations: Perfection Limit minimal per-round terminating probability 3.A special distribution: allows to construct a correlated  -equilibrium. Adaptations: unrevealing, universal device

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 26 Summary Reductions

27 Ramsey Theorem (1930)  A finite set of colors  Each 2 integers (k,n) are colored by c(k,n)  There is an infinite sequence of integers k 1 <k 2 <k 3 <… such that: c(k 1,k 2 ) =c(k i,k j ) for all i<j 0123456789101112 k1k1

28 Stochastic Variation of Ramsey Theorem (Shmaya & Solan, 04)  Coloring each finite sub-tree.  There is an increasing sequence of stopping times:  1 1-  Low probability 11

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 29 Summary Reductions

30 Equilibrium Construction  Each finite tree is colored according to: Whether it has a non-absorbing perfect equilibrium, an absorbing perfect equilibrium, or a special distribution The equilibrium payoff The maximal payoffs when a player stops alone  If c implies that each game on finite tree has a perfect equilibrium, concatenate the equilibria to obtain an approximate perfect equilibrium of G

31 Equilibrium Construction  Last case: c implies that a special distribution exists  This allow to construct an approximate unrevealing perfect correlated equilibrium with a universal correlation device An adaptation of the protocol of Solan and Vohra (01)

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 32 Summary Reductions

33 Summary and Future Research  Summary: every multi-player stopping game admits an approximate normal-form uniform perfect correlated equilibrium with a universal correlation device  Future research: Using this notion of equilibrium in the study of other dynamic games Structure of uniform perfect correlated equilibrium payoffs in specific applications

34 Questions & Comments?  Y. Heller (2008), Perfect correlated equilibria in stopping games, mimeo. http://www.tau.ac.il/~helleryu/

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 35 Summary Reductions

36 Reduction to Terminating Games  Proposition: Every game that stops immediately admits a ( ,  )- unrevealing perfect correlated (   )-equilibrium with a universal correlation device Every stopping game admits the same kind of (   )-equilibrium

37 Proof Outline  Induction on the number of players  Given a stopping game G, we define an auxiliary terminating game G’: The payoff to I \ S when a coalition S stops is the equilibrium payoff in the induced stopping game G’ admits an unrevealing perfect correlated (   )-equilibrium with a universal correlation device Concatenation gives such an equilibrium in G

38 Tree-like Games  Shmaya & Solan (2002) showed that any stopping game can be approximated by a tree-like stopping game, with the same set of approximate equilibria Small perturbations of the payoffs don’t change the set of approximate equilibria we can assume that the payoff process has a finite range Each set F n  F n can be identified with a node in a tree

Tree-like Games – Shmaya & Solan’s Proof Outline  k th partition: Discretization of the game: Depth: k Precision:     Refinement of all previous partitions Defines the k th approximating game on a tree  The game on finite tree that begins on m and ends on l will be played on the m+l approximating game

40 Tree-like Games F4F4F4F4 F1F1 F2F2 F4F4

41 Deep Enough in the Tree FnFn v1v1 v1v1 v1v1 v1v1 v 1, v 3, v 5 occur infinitely often, all other v  V do not occur at all v1v1 v1v1 v1v1 G (F n ): The induced game that begins at the node F n

42 Lemma - Induced Games  Let: G - a terminating game,  - a stopping time Every induced game G (F n ), where F n is in the range of , admits an unrevealing perfect correlated (   )- equilibrium with a universal correlation device G admits the same kind of (C·   )-equilibrium Corollary: We can assume to be “deep enough”

Proof Outline  Until   the players follow an equilibrium in a finite stopping game with absorbing states {F  } with payoffs {x F   equilibria payoffs of G (F  )  After  players follow (   )-equilibrium of G (F   )  Relying on that the equilibrium is unrevealing and with a universal correlation device  Illustration…. 43

Proof Outline FF 44 x1x1 x2x2 x3x3 V(|I|,  ) - universal correlation device x4x4 x5x5 x6x6 x7x7

45 Games on Finite Trees  g i : maximal payoff player i can get by stopping alone  The special distribution  over (nodes · players): A stopping player i and a node with maximal payoff (R i i,n =g i ) The distribution gives each player i at-least g i Each stopping player has a punisher j that stops when R j i,n <g i Allows to construct a correlated  -equilibrium

46 Stationary Absorbing Equilibrium: Adaptations  Perfection - using a perturbed tree with  probability to ignore players’ requests to stop  Limiting the minimal per-round terminating probability  (adapting the methods of Shmaya & Solan, 2004) If there is a player i with a payoff below g i, then  can’t be too small or player i stops when his payoff is g i Otherwise either case 3 applies, or there is a node where at-least 2 players stop with a non-negligible probability Recursive trimming of such nodes gives the needed limit

47  Last case: c implies that a special distribution  exists  Let   i k be the k-th time that player i’s maximal payoff occur with the requirement   i k >   j k-1 for all i, j Using the fact that we are “deep enough” in the tree  An approximate unrevealing perfect correlated equilibrium with a universal correlation device is constructed as follows… Equilibrium Construction: Protocol Description `

48 Equilibrium Construction: Protocol Description  A quitter i’ is secretly chosen according to the special distribution  A number l’ is chosen uniformly in {1,T’}  i’ receives the signal l’  A number l is chosen uniformly in {l’+1,l’+T} 1<<T<<T‘  The punisher of i’ receives the signal l  Each other player receives a signal l+1  Approximate unrevealing perfect correlated equilibrium: each player stops at  l (when l is his signal) modulo 1+T+T’

Introduction: Stopping games perfect correlated (  )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 49 Summary Reductions

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