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

Extensive Games with Perfect Information – p. 3/21 Example 2: Battle of the Sexes

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

Extensive Games with Perfect Information – p. 4/21 Example 3: Matching Pennies

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

Extensive Games with Perfect Information – p. 5/21 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.

Extensive Games with Perfect Information – p. 6/21 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 ,

outcome 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 on the edges.

Extensive Games with Perfect Information – p. 7/21 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 6∈ 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.

Extensive Games with Perfect Information – p. 8/21 Example of Strategies: Matching Pennies

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

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

Extensive Games with Perfect Information – p. 9/21 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.

Extensive Games with Perfect Information – p. 10/21 Example: Matching Pennies

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

Extensive Games with Perfect Information – p. 11/21 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.

Extensive Games with Perfect Information – p. 12/21 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.

Extensive Games with Perfect Information – p. 13/21 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 x 2 A R (x, 100 − x) (0, 0)

Extensive Games with Perfect Information – p. 14/21 Nash Equilibria in the Ultimatum Game

1 x 2 A R (x, 100 − x) (0, 0) 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∗).

Extensive Games with Perfect Information – p. 15/21 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

s2 ...... x1 x1, 100 − x1 ...... ∗ ∗ ∗ x x1, 100 − x1 ...... x2 0, 0 ...... Conclusion. The notion of a Nash equilibrium is not informative here.

Extensive Games with Perfect Information – p. 16/21 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 v v determines his strategy si in EG .

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

Extensive Games with Perfect Information – p. 17/21 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 ); od Note. This process generates a set of joint strategies. Multiple joint strategies may arise due to the second choose statement.

Extensive Games with Perfect Information – p. 18/21 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.

Extensive Games with Perfect Information – p. 19/21 Example: Ultimatum Game

1 x 2 A R (x, 100 − x) (0, 0) 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 =6 100 then A else R fi), with the outcome (99, 1).

Extensive Games with Perfect Information – p. 20/21 Example: the Centipede Game

(Rosenthal, 1981) 1aC 2aC 1bC 2bC 1c C 2c C (4, 4) S S S S S S

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