Extensive Games with Perfect Information Krzysztof R. Apt CWI, - - PowerPoint PPT Presentation

Extensive Games with Perfect Information

Krzysztof R. Apt

CWI, Amsterdam, the Netherlands, University of Amsterdam

Extensive Games with Perfect Information – p. 1/21

Overview

Examples. Definitions. Nash equilibrium. Subgame perfect equilibrium. Backward induction.

Extensive Games with Perfect Information – p. 2/21

Example 1: Prisoner’s Dilemma

1 2 (2,2) C (0,3) D C 2 (3,0) C (1,1) D D

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Example 2: Battle of the Sexes

1 2 (2,1) F (0,0) B F 2 (0,0) F (1,2) B B

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Example 3: Matching Pennies

1 2 (1,-1) H (-1,1) T H 2 (-1,1) H (1,-1) T T

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Discussion

These are examples of two-player games with two stages. In general there may be more players and more stages. We limit ourselves to the games with finitely many stages (games with finite horizon) and such that at each stage exactly one player proceeds.

• Note. At each stage a player can have infinitely many

choices. We assume here perfect information: each player knows the previous moves.

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Extensive Game: Definition

Extensive game for n ≥ 1 players: game tree: a finite depth tree T := (V, E) with a turn function D : V \ Z → {1, . . ., n}, where Z is the set of leaves of T,

• utcome function oi : Z → R, for each player i.

We denote it by (T, D, o1, . . ., on). Given v ∈ V \ Z we call {w | (v, w) ∈ E} the set of actions available to player D(v) at node v. Sometimes we identify the actions with the labels put

• n the edges.

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Strategies

Consider an extensive game EG := (T, D, o1, . . ., on). Let Ni := {v ∈ V | D(v) = i}.

Ni is the set of nodes at which player i takes an action.

Strategy for player i:

si : Ni → V , such that for all v ∈ Ni, (v, si(v)) ∈ E.

Joint strategy: s = (s1, . . ., sn). It assigns a unique edge to every node in V \ Z. To each joint strategy s there corresponds a finite path

path(s) := (v1, . . ., vh) in T defined inductively: v1 is the root of T,

if vk ∈ Z, then vk+1 := si(vk), where D(vk) = i. When each player i selects si we call (o1(z), . . ., on(z)), where z is the last element of path(s), the outcome of EG.

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Example of Strategies: Matching Pennies

1 2 (1,-1) H (-1,1) T H 2 (-1,1) H (1,-1) T T Strategies for player 1: H, T. Strategies for player 2: HH, HT, TH, TT. Thick lines correspond with the joint strategy (T,HH).

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Strategic Forms

With each extensive game EG := (T, D, o1, . . ., on) we associate a strategic game G := (S1, . . ., Sn, p1, . . ., pn) defined as follows:

Si is the set of strategies of player i in EG, pi(s) := oi(z), where z is the last element of path(s). G is called the strategic form of EG. s is called a Nash equilibrium of EG if it is a Nash

equilibrium of G.

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Example: Matching Pennies

1 2 (1,-1) H (-1,1) T H 2 (-1,1) H (1,-1) T T Strategic form

HH HT TH TT H 1, −1 1, −1 −1, 1 −1, 1 T −1, 1 1, −1 −1, 1 1, −1

• Note. Two Nash equilibria: (H, TH) and (T, TH).

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Win or Lose Games

A two-player extensive game is called a win or lose game if the only possible outcomes are (1, −1) and

(−1, 1). si is called a winning strategy of player i in a win or lose

game EG if

∀s−i ∈ S−ipi(si, s−i) = 1,

where (S1, S2, p1, p2) is the strategic form of EG. Theorem (Zermelo, 1913) In every win or lose game one of the players has a winning strategy.

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Proof of Zermelo’s Theorem

Theorem In every win or lose game one of the players has a winning strategy. We can assume that the players alternate their moves. We can extend all the paths in the game so that all paths in T are of the same depth, say 2k. Let W denote the sentence “player 1 wins after 2k stages”. Then the formula

φ1 := ∃x1∀y1. . .∃xk∀ykW

denotes “player 1 has a winning strategy” and

φ2 := ∀x1∃y1. . .∀xk∃yk¬W

denotes “player 2 has a winning strategy”. But ¬φ1 ≡ φ2, i.e., φ1 ∨ φ2 holds.

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Example: Ultimatum Game

Player 1 claims x ∈ {0, 1, . . ., 100}. Player 2 accepts - the outcome is then (x, 100 − x), or rejects - the outcome is then (0, 0). For each x ∈ {0, 1, . . ., 100} the root 1 has the subtree 1 2

(x, 100 − x)

A

(0, 0)

R

x

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Nash Equilibria in the Ultimatum Game

1 2

(x, 100 − x)

A

(0, 0)

R

x

• Note. For each x ∈ {0, 1, . . ., 100} there is a Nash

equilibrium with the outcome (x, 100 − x).

• Proof. Take x∗ ∈ {0, 1, . . ., 100}.

Strategy for player 1: x∗. Strategy for player 2: if x ≤ x∗ then A else R fi. This is a Nash equilibrium with the outcome (x∗, 100 − x∗).

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Example: Ultimatum Game, ctd

Illustration. Strategy for player 1: x∗. Strategy for player 2: s2 := A if x ≤ x∗ and R otherwise. Consider two deviations of player 1, x1 < x∗ < x2.

s2 . . . . . . . . . x1 x1, 100 − x1 . . . . . . x∗ x∗

1, 100 − x∗ 1

. . . . . . x2 0, 0 . . . . . . . . . . . . . . .

• Conclusion. The notion of a Nash equilibrium is not

informative here.

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Subgames

Consider EG := (T, D, o1, . . ., on). We define the subgame of EG rooted at node v of T,

EGv, as expected.

• Note. Each strategy si of player i in EG uniquely

determines his strategy sv

i in EGv.

(s1, . . ., sn) is called a subgame perfect equilibrium in EG if for each node v of T (sv

1, . . ., sv n) is a Nash

equilibrium in EGv. Informally: s is subgame perfect equilibrium in EG if it induces a Nash equilibrium in every subgame of EG.

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Backward Induction

Given a tree (V, E) and v ∈ V , let

desc(v) := {w | (v, w) ∈ E}.

Fix a finite extensive game EG := ((V, E), D, o1, . . ., on). Backward induction algorithm while |V | > 1 do choose v ∈ V such that all its descendants are leaves;

i := D(v);

choose w ∈ desc(v) such that oi(w) is maximal;

si(v) := w;

for j ∈ {1, . . ., n} do oj(v) := oj(w) od;

V := V \ desc(v); E := E ∩ (V × V );

• d
• Note. This process generates a set of joint strategies.

Multiple joint strategies may arise due to the second choose statement.

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Kuhn and Selten Theorems

Theorem (Kuhn, 1950) Every finite extensive game (with perfect information) has a Nash equilibrium. Theorem (Selten, 1965) Every finite extensive game (with perfect information) has a subgame perfect equilibrium.

• Proof. A stronger claim holds:

A joint strategy is a subgame perfect equilibrium iff it can be generated by the backward induction algorithm.

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Example: Ultimatum Game

1 2

(x, 100 − x)

A

(0, 0)

R

x

Player 2 has two best responses to the strategy 100: A and R.

• Note. There are two subgame perfect equilibria:

(100, always A), with the outcome (100, 0), (99, if x = 100 then A else R fi), with the outcome

(99, 1).

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Example: the Centipede Game

(Rosenthal, 1981)

S S S S S S C C C C C C

1a 1b 1c 2a 2b 2c

(1, 1) (0, 3) (2, 2) (1, 4) (3, 3) (2, 5) (4, 4)

General rule: Initial situation: (1, 1). If a player continues he loses 1 and the opponent gains 2.

• Note. Backward induction shows that in the unique

subgame perfect equilibrium each player selects at each node S. So the outcome of the game is (1, 1).

(1, 1) is also the outcome of the game in each Nash

equilibrium.

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