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1
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
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
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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/
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