Stochastic Games

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# Stochastic Games - PowerPoint PPT Presentation

Stochastic Games. Mr Sujit P Gujar. e-Enterprise Lab Computer Science and Automation IISc, Bangalore. Agenda. Stochastic Game Special Class of Stochastic Games Analysis : Shapley’s Result. Applications. Repeated Game.

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### Stochastic Games

Mr Sujit P Gujar.

e-Enterprise Lab

Computer Science and Automation

IISc, Bangalore.

Agenda
• Stochastic Game
• Special Class of Stochastic Games
• Analysis : Shapley’s Result.
• Applications

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Repeated Game
• When players interact by playing a similar stage game (such as the prisoner's dilemma) numerous times, the game is called a repeated game.

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Stochastic Game
• Stochastic game is repeated game with probabilistic/stochastic transitions.
• There are different states of a game.
• Transition probabilities depend upon actions of players.
• Two player stochastic game : 2 and 1/2 player game.

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First Iteration

subgame

Second

Iteration

Repeated Prisoner’s Dilemma
• Consider Game tree for PD repeated twice.

Assume each player has the same two options at each info set: {C,D}

1

2

1

1

1

1

2

2

2

2

What is Player 1’s strategy set?(Cross product of all choice sets at all information sets…)

{C,D} x {C,D} x {C,D} x {C,D} x {C,D}

25 = 32 possible strategies

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Issues in Analyzing Repeated Games
• How to we solve infinitely repeated games?
• Strategies are infinite in number.
• Need to compare sums of infinite streams of payoffs

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Stochastic Game : The Big Match
• Every day player 2chooses a number, 0 or 1
• Player 1 tries to predict it. Wins a point if he is correct.
• This continues as long as player 1 predicts 0.
• But if he ever predicts 1, all future choices for both players are required to be the same as that day's choices.

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The Big Match
• S = {0,1*,2*} : State space.
• s0={0,1} s1={0} s2={1}
• P02 =
• N = {1,2}
• P00 =
• A = Payoff Matrix

=

• P01 =

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The "Big-Match" game is introduced by Gillette (1957) as a difficult example.
• The Big Match

David Blackwell; T. S. Ferguson

The Annals of Mathematical Statistics, Vol. 39, No. 1. (Feb., 1968), pp. 159-163.

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Scenario

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Stationary Strategies
• Enumerating all pure and mixed strategies is cumbersome and redundant.
• Behavior strategies those which specify a player the same probabilities for his choices every time the same position is reached by whatever route.
• x = (x1,x2,…,xN) each xk = (xk1, xk2,…, xkmk)

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Notation
• Given a matrix game B,
• val[B] = minimax value to the first player.
• X[B] = The set of optimal strategies for first player.
• Y[B] = The set of optimal strategies for second player.
• It can be shown, (B and C having same dimensions)

|val[B] - val[C]| ≤ max |bij - cij|

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When we start in position k, we obtain a particular game,
• We will refer stochastic game as,

Define,

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Shapley’s1 Results

1L.S. Shapley, Stochastic Games. PNAS 39(1953) 1095-1100

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Let, denote the collection of games whose pure strategies are the stationary strategies of . The payoff function of these new games must satisfy,

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Shapley’s Result,

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Applications
• 1When N = 1,
• By setting all skij = s > 0, we get model of infinitely repeated game with future payments are discounted by a factor = (1-s).
• If we set nk = 1 for all k, the result is “dynamic programming model”.

1von Neumann J. , Ergennise eines Math, Kolloquims, 8 73-83 (1937)

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Example
• Consider the game with N = 1,
• A =
• P2 =
• P1 =
• x=(0.61,0.39)
• y=(0.39, 0.61)
• x=(0.6,0.4)
• y=(0.4, 0.6)

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Thank You!!

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