Matrix%20Games

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Matrix Games

• Mahesh Arumugam
• Borzoo Bonakdarpour
• Ali Ebnenasir
• CSE 960 Selected Topics in Algorithms and
  Complexity
• Instructor Dr. Torng

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Outline

• Basic concepts
• Problem statement
• LP Formulation of Matrix Games
• Minimax Theorem
• Gambling
• Bluffing and Underbidding

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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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ExampleThe 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

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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
A
B
0
2
-3
0
1,1
1,2
-2
0
0
3
2,1
3
0
0
-4
2,2
0
-3
4
0

• 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

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The Game of Morra - Contd

• 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 c2c3 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 - Contd

• 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

Possible moves of the column player
Possible moves of the row player
1 2 j
n
.
a11
1 2 i . . m
.
aij
.
amn
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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

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LP Formulation of Matrix Games - Contd

• If row player adopts the strategy specified by
  stochastic vector x, he is assured to win
• The objective is to maximize this payoff

s.t.,
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LP Formulation of Matrix Games - Contd

• What is the dual of this problem?

P

• What does this problem formalize?

Column players optimal strategy and the value he
is assured to win if he adopts such a strategy!
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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 xAy 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 xAy 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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GamblingPayoff Matrix and LP Solution

• Consider the optimal solution
• x0, 56/99, 40/99, 0, 0, 2/99, 0, 1/99
• y28/99, 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?

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

• As strategies
• Pass if B bets, pass again
• Pass if B bets, bet again
• Bet
• 3x3x3 pure strategies
• x1x2x3 strategy for A instructing him to follow
  line xj when holding j

• Bs strategies
• Pass no matter what A did
• If A passes, pass if A bets, bet
• If A passes, bet if A bets, pass
• Bet no matter what A did
• 4x4x4 pure strategies
• y1y2y3 strategy for B

Payoff matrix size 27x64!
Payoff matrix size 8x4!
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
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Bluffing, Underbidding Payoff Matrix and LP
Solution

• 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

• 114 124 314 324
• 11 2 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

Holding 1 BLUFF
A is allowed to bet 1/6th times! B is allowed to
bet 1/3rd times!
Holding 3 UNDERBID
A is allowed to pass 1/2 times!
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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 lt minj ?im aij xi
• Now, we show
• miny xAy gt minj ?im aij xi
• Let t minj ?im aij xi , thus we have
• xAy ?jn yj (?im aij xi) gt
  ?jn yj t t