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Sequential Correlated Equilibria in Stopping Games

Yuval HellerTel-Aviv University

(Part of my Ph.D. thesis supervised by Eilon Solan)

http://www.tau.ac.il/~helleryu/

Stony-Brook

July 2010

Contents

Introduction Motivating example

Solution concept

Simplifying assumptions

Model: stopping games

Main result: equilibrium existence Proof’s sketch

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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

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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

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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

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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

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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

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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

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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 Ti < 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”

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Multi-Player Stopping Games (2)

Player who acted Ti 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): Ti=1 for all players (by induction )

Each player has a single “stopping” action Game ends as soon as any player stops

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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 (2nd price all-pay, Krishna & Morgan, 97) Lobbying (Bulow & Klemperer, 01) Conflict among animals (Nalebuff & Riley, 85) Ti>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

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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

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Proof’s Sketch

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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.)

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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

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Concatenating Equilibria

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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: k1<k2<k3<… such that: c(k1,k2) =c(ki,kj)

for all i<j

0 1 2 3 4 5 6 7 8 9 10 11 12k1

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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<2<3<…, s.t. Pr(c(1,2) =c(,) ….)>1-

Low probability

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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

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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

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Questions & Comments?

Yuval Heller

http://www.tau.ac.il/~helleryu/