Zero-sum Games

• The Essentials of a Game• Extensive Game• Matrix Game• Dominant Strategies• Prudent Strategies• Solving the Zero-sum Game• The Minimax Theorem

The Essentials of a Game1. Players: We require at least 2 players (Players choose actions

and receive payoffs.)

2. Actions: Player i chooses from a finite set of actions, S = {s1,s2,…..,sn}. Player j chooses from a finite set of actions T = {t1,t2,……,tm}.

3. Payoffs: We define Pi(s,t) as the payoff to player i, if Player i chooses s and player j chooses t. We require that Pi(s,t) + Pj(s,t) = 0 for all combinations of s and t.

4. Information: What players know (believe) when choosing actions.

ZERO-SUM

The Essentials of a Game

4. Information: What players know (believe) when choosing actions.

Perfect Information: Players know

• their own payoffs • other player(s) payoffs • the history of the game, including other(s) current action*

*Actions are sequential (e.g., chess, tic-tac-toe).

Common Knowledge

Extensive GamePlayer 1 chooses a = {1, 2 or 3} Player 2 b = {1 or 2} Player 1 c = {1, 2 or 3}

Payoffs = a2 + b2 + c2 if /4 leaves remainder of 0 or 1. -(a2 + b2 + c2) if /4 leaves remainder of 2 or 3. Player1’sdecision nodes

-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.

“Square the Diagonal”(Rapoport: 48-9)

Player 2’sdecision nodes

1 32

1 21

23

Extensive GameHow should the game be played?Solution: a set of “advisable” strategies, one for each player.Strategy: a complete plan of action for every possible decision

node of the game, including nodes that could only be reached by a mistake at an earlier node.

Player1‘s advisable Strategy in red

-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.

1 32

1 21

23

Start at the final decision nodes (in red) Backwards-induction

Extensive GameHow should the game be played?Solution: a set of “advisable” strategies, one for each player.Strategy: a complete plan of action for every possible decision

node of the game, including nodes that could only be reached by a mistake at an earlier node.

Player1‘s advisable Strategy in red

-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.

Player2’s advisable strategy in green

1 32

1 21

23

Player1’s advisable strategy in red

Extensive GameHow should the game be played?If both player’s choose their advisable (prudent) strategies, Player1 will start with 2, Player2 will choose 1, then Player1 will choose 2. The outcome will be 9 for Player1 (-9 for Player2). If a player makes a mistake, or deviates, her payoff will be less.

-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.

1 32

1 21

23

Extensive GameA Clarification: Rapoport (pp. 49-53) claims Player 1 has 27 strategies. However, if we consider inconsistent strategies, the actual number of strategies available to Player 1 is 37 = 2187.

An inconsistent strategy includes actions at decision nodes that would not be reached by correct implementation at earlier nodes, i.e., could only be reached by mistake.

Since we can think of a strategy as a set of instructions (or program) given to an agent or referee (or machine) to implement, a complete strategy must include instructions for what to do after a mistake is made. This greatly expands the number of strategies available, though the essence of Rapoport’s analysis is correct.

Extensive GameComplete Information: Players know their own payoffs;

other player(s) payoffs; history of the game excluding other(s) current action*

*Actions are simultaneous

-3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22GAME 1.

1 32

1 21

23

Information Sets

Matrix Game

-3 -6-6 9

-11 -14-6 99 12

-14 17-11 -14-14 17-19 -22

T1 T2

Also called “Normal Form” or “Strategic Game”

Solution = {S22, T1}

S11

S12

S13

S21

S22

S23

S31

S32

S33

Dominant StrategiesDefinition

Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s).

T1 T2 T3 T1 T2 T3

-3 0 -10

-1 5 2

-2 -4 0

-3 0 1

-1 5 2

-2 2 0

S1

S2

S3

S1

S2

S3

Dominant StrategiesDefinition

Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s).

T1 T2 T3 T1 T2 T3

Sure Thing Principle: If you have a dominant strategy, use it!

-3 0 -10

-1 5 2

-2 -4 0

-3 0 1

-1 5 2

-2 2 0

S1

S2

S3

S1

S2

S3

Prudent Strategies

T1 T2 T3

Player 1’s worst payoffs for each strategy are in red.

-3 1 -20

-1 5 2

-2 -4 15

S1

S2

S3

Definitions

Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i.

Prudent Strategies

T1 T2 T3

Player 2’s worst payoffs for each strategy are in green.

-3 1 -20

-1 5 2

-2 -4 15

S1

S2

S3

Definitions

Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i.

Prudent Strategies

T1 T2 T3

-3 1 -20

-1 5 2

-2 -4 15

S1

S2

S3

Definitions

Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply maxsmintP(s,t) for player i.

Saddlepoint: A set of prudent strategies (one for each player), s. t. (s’, t’) is a saddlepoint, iff maxmin = minmax.

We call the solution {S2, T1} a saddlepoint

Prudent Strategies

-3 1 -20

-1 5 2

-2 -4 15

S1

S2

S3

Saddlepoint: A set of prudent strategies (one for each player), s. t. (s’, t’) is a saddlepoint, iff

maxmin = minmax.

Mixed Strategies

Left Right

L R L R

-2 4 2 -1

Player 1

Player 2

GAME 2: Button-Button

Player 1 hides a button in his Left or Right hand.

Player 2 observes Player 1’s choice and then picks either Left or Right.

Draw the game in matrix form.

Mixed Strategies

Left Right

L R L R

-2 4 2 -1

Player 1

Player 2

GAME 2: Button-Button

Player 1 has 2 strategies;Player 2 has 4 strategies:

-2 4 -2 4

2 -1 -1 2

L

R

LL RR LR RL

Mixed Strategies

Left Right

L R L R

-2 4 2 -1

Player 1

Player 2

GAME 2: Button-Button

The game can be solve by backwards-induction. Player 2 will …

-2 4 -2 4

2 -1 -1 2

L

R

LL RR LR RL

Mixed Strategies

Left Right

L R L R

-2 4 2 -1

Player 1

Player 2

GAME 2: Button-Button

The game can be solve by backwards-induction. … therefore, Player 1 will:

-2 4 -2 4

2 -1 -1 2

L

R

LL RR LR RL

Mixed Strategies

Left Right

L R L R

-2 4 2 -1

Player 1

Player 2

-2 4

2 -1

L R

L

R

GAME 2: Button-Button

What would happen if Player 2 cannot observe Player 1’s choice?

Solving the Zero-sum Game

GAME 2.

-2 4

2 -1

Definition

Mixed Strategy: A mixed strategy for player i is a probability distribution over all strategies available to player i.

Let (p, 1-p) = prob. Player I chooses L, R.(q, 1-q) = prob. Player 2 chooses L, R.

L R

L

R

Solving the Zero-sum Game

GAME 2.

-2 4

2 -1

Then Player 1’s expected payoffs are: EP(L) = -2(p) + 2(1-p) = 2 – 4p EP(R) = 4(p) – 1(1-p) = 5p – 1

L R

L

R

(p)

(1-p)

(q) (1-q)0 1 p

EP(L) = 2 – 4p

EP(R) = 5p – 1

EP

p*=1/3

2

-1

4

-2

Solving the Zero-sum Game

GAME 2.

-2 4

2 -1

Player 2’s expected payoffs are:

EP(L) = 2(q) – 4(1-q) = 6q – 4 EP(R) = -2(q) + 1(1-q) = -3q + 1

EP(L) = EP(R) => q* = 5/9

L R

L

R

(p)

(1-p)

(q) (1-q)

Solving the Zero-sum GamePlayer 1

EP(L) = -2(p) + 2(1-p) = 2 – 4p EP(R) = 4(p) – 1(1-p) = 5p – 1

0 p 1 q

-EP2

p*=1/3

2

-1

4

-2

2/3 = EP1* = - EP2* =-2/3

This is the

Value

of the game.

EP1

-4

2

-2

2

q*= 5/9

Player 2

EP(L) = 2(q) – 4(1-q) = 6q – 4 EP(R) = -2(q) + 1(1-q) = -3q + 1

Solving the Zero-sum Game

GAME 3.

-2 4

2 -1

Then Player 1’s expected payoffs are:

EP(T1) = -2(p) + 2(1-p) EP(T2) = 4(p) – 1(1-p)

EP(T1) = EP(T2) => p* = 1/3

And Player 2’s expected payoffs are:

(V)alue = 2/3

L R

L

R

(p)

(1-p)

(q) (1-q)

(Security) Value: the expected payoff when both (all) players play prudent strategies.

Any deviation by an opponent leads to an equal or greater payoff.

The Minimax Theorem

Von Neumann (1928)

Every zero sum game has a saddlepoint (in pure or mixed strategies), s.t., there exists a unique value, i.e., an outcome of the game where

maxmin = minmax.