Matrix Games

Mahesh ArumugamBorzoo BonakdarpourAli Ebnenasir

CSE 960: Selected Topics in Algorithms and ComplexityInstructor: Dr. Torng

2

Outline

• Basic concepts

• Problem statement

• LP Formulation of Matrix Games

• Minimax Theorem

• Gambling

• Bluffing and Underbidding

3

Basic Concepts

• Game: A description of strategic interaction between rationale parties based on a set of rules

• Rules: Constraints on the set of actions that each party can take and the players’ interest

• Finite Game: Set of actions of each player is finite

• Two-Player Game: There exist only two players

[OR94] Osborne and Rubinstein, A Course in Game Theory, MIT press, 1994.

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Example:The Game of Morra

• Rule: – Each player hides one or two francs, and– Tries to guess how many francs the other player has

hidden

• Payoff:– If only one player guesses correctly

• he wins the total amount of hidden money

– Otherwise, the result is a draw

5

The Game of Morra: Pure Strategies

• Possible courses of action for each player– Hide one, guess one [1, 1]– Hide one, guess two [1, 2]– Hide two, guess one [2, 1]– Hide two, guess two [2, 2]

• Pure strategy: a course of action– Denoted [x,y]; i.e., hide x, guess y

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The Game of Morra: Payoff Matrix

[1,1][1,2][2,1][2,2]

[1,1] [1,2] [2,1] [2,2]

0

-2

30

2 -3 0

0

0-3

0

04 0

-4

3

A B

xi – probability that row i is selected by row player

yj – relative frequency with which column j is selected

by column player– X and Y are stochastic vectors

y1 y2 y3 y4y = [ ]

x1

x2

x3

x4

x =

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The Game of Morra - Cont’d

• A only plays [1,2] or [2,1] with probability 0.5• B plays

– [1,1] , [1,2], [2,1], [2,2] in c1, c2, c3, c4 rounds • c1+ c2+c3 +c4 = N, where N is total number of rounds

• Record of the game – In c1/2 rounds, A played [1,2] and B played [1,1]: A losing 2 francs– In c1/2 rounds, A played [2,1] and B played [1,1]: A winning 3 francs– In c4/2 rounds, A played [1,2] and B played [2,2]: A winning 3 francs– In c4/2 rounds, A played [2,1] and B played [2,2]: A losing 4 francs– Other rounds, result in a draw

• Total winning of A : (c1 – c4)/2 francs

What if the roles of A and B are swapped?

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Basic Concepts - Cont’d

• Round: a course of actions in which each player moves once

• Payoff: the value gained by a player in a round• The Payoff Matrix defines a game for two players• Zero-sum game: The sum of the average payoffs of the

two players is 0

a11

aij…….

…….

……. amn

Possible movesof the row player

12

i..m

Possible movesof the column player

1 2 … j … n

The resulting payoffof the row player

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Problem Statement

Given the payoff matrix A = [aij ],

– identify a mixture of moves of the row player where the average payoff per round is optimal no matter what moves the column player takes

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LP Formulation of Matrix Games

xi – probability that row i is selected by row player

yj – relative frequency with which column j is selected

by column player– X and Y are stochastic vectors

• Average payoff to the row player in each round

m

i

n

jjiij yxa

1 1

xAyor

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LP Formulation of Matrix Games - Cont’d

• If row player adopts the strategy specified by stochastic vector x, he is assured to win

=

• The objective is to maximize this payoff

xAyy

min

m

iiij

jxa

1

min

m

iiij

jxa

1

minmaximize

11

m

iix

m) , 2, 1, (i0 ix

s.t.,

zmaximize

s.t., n) , 2, 1, (j01

m

iiij xaz

11

m

iix

m) , 2, 1, (i0 ix

or

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LP Formulation of Matrix Games - Cont’d

• What is the dual of this problem?

• What does this problem formalize?

Column player’s optimal strategy and the value he is assured to win if he adopts such a strategy!

zmaximize

s.t., n) , 2, 1, (j01

m

iiij xaz

11

m

iix

m) , 2, 1, (i0 ix

P wminimize

s.t., m) , 2, 1, (i01

n

jjij yaw

11

n

jjy

n) , 2, 1, (j0 jy

D

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Minimax Theorem

For every m n matrix A there is a stochastic row vector x* of length m and a stochastic column vector y* of length n such that

min x*Ay = max xAy*

with the minimum taken over all stochastic column vectors y of length n and maximum taken over all stochastic row vectors x of length m.

Value of game

In a game, v = min x*Ay = max xAy* is called the value of that game.

What are the implications of this theorem?

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Ready for Gambling?!!

• As long as a player adopts an optimal strategy, the player can reveal it to the opponent

• Example: (The Game of Morra)– column player announces his/her guess – row player announces his/her guess either independent of the

opponent or adjust his/her guess based on the extra information

– Additional pure strategies for row player• Hide 1, make the same guess [1, S]• Hide 1, make a different guess [1, D]• Hide 2, make the same guess [2, S]• Hide 2, make a different guess [2, D]

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Gambling:Payoff Matrix and LP Solution

Consider the optimal solutionx=[0, 56/99, 40/99, 0, 0, 2/99, 0, 1/99]y=[28/99, 30/99, 21/99, 20/99]Game value = 4/99

- row player is assured to win at least this amount on the average

- column player is assured to lose no more than this amount on the average

Do you think this game is fair?What does this suggest?

4400

0033

0022

3300

0430

4003

3002

0320

],2[

],2[

],1[

],1[

]2,2[

]1,2[

]2,1[

]1,1[

D

S

D

S

[1,1] [1,2] [2,1] [2,2]

Revealing the guess does not hurt the prospects for the column player!!

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How about Bluffing or Underbidding?

• Are bluffing or underbidding rational strategies?

• Example: (Game invented by H. W. Kuhn)

– 2 players, deck of cards numbered 1, 2, or 3– Each player bets or passes in every round– Play terminates when

• Bet is answered by bet; payoff 2 to player holding higher card• Pass is answered by pass; payoff 1 to player holding higher

card• Bet is answered by pass; payoff 1 to the player who bets

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Bluffing, Underbidding: Pure Strategies

• A’s strategies1. Pass; if B bets, pass again2. Pass; if B bets, bet again3. Bet

3x3x3 pure strategies

• x1x2x3 – strategy for A instructing him to follow line xj when holding j

• B’s strategies1. Pass no matter what A did2. If A passes, pass; if A bets, bet3. If A passes, bet; if A bets, pass4. Bet no matter what A did

4x4x4 pure strategies

• y1y2y3 – strategy for B

Payoff matrix size: 27x64!

Holding 1: A – refrain line 2; B – refrain lines 2 and 4; Holding 3: A – refrain line 1; B – refrain lines 1, 2 and 3;Holding 2: choose to pass in the first round; lines 1 or 2

Payoff matrix size: 8x4!

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Bluffing, Underbidding: Payoff Matrix and LP Solution

114 124 314 324112 0 0 -1/6 -1/6

113 0 1/6 -1/3 -1/6

122 -1/6 -1/6 1/6 1/6

123 -1/6 0 0 1/6

312 1/6 -1/3 0 -1/2

313 1/6 -1/6 -1/6 -1/2

322 0 -1/2 1/3 -1/6

323 0 -1/3 1/6 -1/6

Consider the optimal solution A: [1/3, 0, 0, 1/2, 1/6, 0, 0, 0]B: [2/3, 0, 0, 1/3]Game Value = -1/18

Holding 1: BLUFF

A is allowed to bet 1/6th times!

B is allowed to bet 1/3rd times!

A is allowed to pass 1/2 times!

Holding 3: UNDERBID

Thank U!

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LP Formulation of Matrix Games: Identity (15.1)

miny xAy = minj im aij xi

– It is trivial that miny xAy <= minj i

m aij xi

– Now, we show miny xAy >= minj i

m aij xi

– Let t = minj im aij xi , thus we have

xAy = jn yj (i

m aij xi) >= jn yj t = t