# Game Theoretic Approach in Computer Science CS3150, Fall 2002

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Game Theoretic Approach in
Computer Science
CS3150, Fall 2002

Introduction to Game Theory
Patchrawat Uthaisombut
University of Pittsburgh

1
Common Knowledge
It

I know it         You know it

I know that you know it        You know that I know it

I know that you know
that I know it.                         It
I
You
…              to know

2
Rationality

• Aware of alternatives
• Has clear preferences among outcomes;
know one’s own value system.
• Doesn’t mean ethical, moral, sensible, selfish
• Value system is incorporated into payoff
function.
• Perfectly good in determining the best
strategy to maximize their payoff.

3
Situation
Pungkang
or
Samkok                  Pungkang
or
Samkok

Apichai
Buncha              4
Payoffs

Buncha

Pungkang     Samkok

Pungkang      2,1            0,0
Apichai
Samkok        0,0            1,2

5
Components of a Game
• Players
• Who is involved?
• Rules
• Who move when?
• What does a player knows when he/she moves?
• What moves are available?
• Outcomes
• For each possible combination of actions by the
players, what’s the outcome of the game.
• Payoffs
• What are the players’ preferences over the possible
outcomes?
6
Components of the Restaurant Game
• Players
• Who is involved?
• Apichai and Buncha
• Rules
• Who move when?
• Apichai and Buncha move simultaneously.
• What does a player knows when he/she moves?
• A and B knows the payoffs matrix but they do not know what
the other’s action is.
• What moves are available?
• Apichai: Pungkang and Samkok
• Buncha: Pungkang and Samkok

7
Components of the Restaurant Game

• Outcomes
• o1: They meet at Pungkang
• o2: They meet at Samkok
• o3: They do not meet.
• Payoffs
• Apichai: o12, o21, o30
• Buncha: o11, o22, o30

8
Properties of all Strategic Games

• Payoff functions are common knowledge
• All players are rational, and this is common
knowledge.
• All players move simultaneously and independently
• All players do not know what the others will do.
• Each player wish to maximize her/his payoff.
• The game is played once.

9
Parameters of a Strategic Game

• A strategic game is a 3-tuple (n,A,u)
• The number of players n.
• For 1<i<n, a set Ai of actions available for
player i.
• For 1<i<n, a payoff function ui:A1…An  R
for player i.

pungkang samkok
pungkang    2,1     0,0
samkok     0,0     1,2
10
Restaurant Game as a Strategic Game

• Players: n = 2                               pungkang samkok
• Player 1 = Apichai         pungkang           2,1        0,0
• Player 2 = Buncha
samkok         0,0        1,2
• Actions:
• A1 = {pungkang, samkok }
• A2 = {pungkang, samkok }
• Payoffs:
•   u1(pungkang,pungkang ) = 2    •   u2(pungkang,pungkang ) = 1
•   u1(pungkang,samkok ) = 0      •   u2(pungkang,samkok ) = 0
•   u1(samkok,pungkang ) = 0      •   u2(samkok,pungkang ) = 0
•   u1(samkok,samkok ) = 1        •   u2(samkok,samkok ) = 2
11
A Play of the Restaurant Game

• The play
• Row player chooses Samkok.
• Column player chooses Samkok.
• (The Outcome)
• They meet at Samkok
• The Payoff                         pungkang samkok
• Row player gets 1.    pungkang     2,1     0,0
• Column player gets 2.
samkok      0,0     1,2
12
Concert Game

• Suppose both Apichai and Buncha are
going to a concert instead of a dinner.
• Both likes Mozart better than Mahler.

Mozart    Mahler
Mozart       2,2       0,0
Mahler       0,0       1,1

13
Movie Game

• Two people go to a movie theatre.

A beautiful
Scream 3
mind
Scream 3          3,2          1,3
A beautiful
2,1            2,2
mind

14
Restaurant Game

• Apichai and Buncha go to a restaurant.

Buncha
Pungkang    Samkok
Pungkang     2,1            0,0
Apichai
Samkok      0,0            1,2

15
The Prisoners’ Dilemma

• The confession of a suspect will be used
against the other.
• If both confess, get a reduced sentence.
• If neither confesses, face only minimum
charge.

Confess    Deny
Confess     -5,-5     0,-10
Deny       -10,0     -1,-1
16
Chicken Game
• Apichai and Buncha dare one another to
drive their cars straight into one another.
Buncha
Swerve      Straight
Swerve        0,0            -1,1
Apichai
Straight     1,-1        -3,-3

17
Matching Pennies

Tail   -1,1   1,-1

18
Notations
• x  Ak
• x is an action or a strategy of player k
• Ak is a set of available actions for player k
• (ai) = (a1, a2,…, an)  A1A2…An = A
• a profile of actions; one action from each player
• (ai) = (X,G,H,L,S)
• (a-k) = (ai) \ ak  A1…Ak-1Ak+1…An = A-k
• actions of everybody except player k
• (a-2) = (X,_,H,L,S)
• (a-k,y) = (a-k)  y
• (a-2,M) = (X,M,H,L,S)
• (a-k,ak) = (ai)
19
Best Responses

• An action x of player k is a best response to
an action profile (a-k) if
• uk(a-k,x) > uk(a-k,y) for all y in Ak.

pungkang samkok                Confess   Deny
pungkang      2,1        0,0       Confess -5,-5    0,-10
samkok        0,0        1,2        Deny   -10,0    -1,-1

20
Dominant Actions
• An action x of player k is a dominant action if
• x is a best response to all (a-k) in A-k.
• That is, uk(a-k,x) > uk(a-k,y) for all y in Ak and any action
profile (a-k) in A-k.
• That is, no matter what the other players do, x is a
strategy for player k that is no worse than any other.

Scream 3         ABM                Confess           Deny
Scream 3    3,2            1,3        Confess -5,-5            0,-10
ABM          2,1          2,2         Deny   -10,0            -1,-1
21
Problems with Dominant Actions

• Sometimes dominant actions do not exist.
• Still need to say something about the game
• Some outcomes are clearly desirable to the
others.

pungkang samkok
pungkang       2,1       0,0
samkok       0,0       1,2

22
Nash Equilibrium
• An action profile (ai) is a Nash equilibrium if
• for every player k, ak is a best response to (a-k)
• that is, for every player k, uk(a-k,ak) > uk(a-k,y) for all y in Ak

• An action x of player k is a dominant action if
• x is a best response to all (a-k).
• that is, uk(a-k,x) > uk(a-k,y) for all y in Ak and any action
profile (a-k) in A-k.

pungkang samkok                    Confess             Deny
pungkang          2,1          0,0        Confess -5,-5              0,-10
samkok           0,0          1,2         Deny   -10,0              -1,-1
23
Strictly Dominated Actions
• An action x of player k is a never-best response or a
strictly dominated action if
• x is not a best response to any action profile (a-k) in A-k
• That is, for any action profile (a-k) in A-k there exist an
action y in Ak such that uk(a-k,x) < uk(a-k,y)
• That is, no matter what the other players do, x is a strategy
for player k that she should never use.

Scream 3         ABM               Confess          Deny
Scream 3    3,2            1,3       Confess -5,-5           0,-10
ABM           2,1         2,2        Deny   -10,0           -1,-1
24
Iterated Elimination of Dominated Actions

• Procedure
• Successively remove a strictly dominated action of a player
from the game table until there are no more strictly dominated
actions
• Removing a dominated action
• Reduce the size of the game
• May make another action dominated
• May make another action dominant
• If there is only 1 outcome remaining,
• the game is said to be dominant solvable.
• that outcome is the unique Nash equilibrium of the game

25
Weakly Dominated Actions
• An action x of player k is a weakly dominated
action if
• for any action profile (a-k) in A-k there exist an
action y in Ak such that uk(a-k,x) < uk(a-k,y) and
• there exist an action profile (a-k) in A-k and an
action y in Ak such that uk(a-k,x) < uk(a-k,y).

Scream 3      ABM               Confess       Deny
Scream 3    3,2         1,3       Confess -5,-5        0,-10
ABM         2,2        2,2        Deny   -10,0        -1,-1
26
Iterated Elimination of Weakly Dominated
Actions
• Procedure
• Same as before except
• Remove weakly dominated actions instead of
strictly dominated actions
• Undesired results
• The remaining cells may depend on the order that
the actions are removed.
• May not yield all Nash equilibria.

27
Best-Response Function
• A set-valued function Bk
• Bk(a-k) = {x  Ak | x is a best response to (a-k) }
• called the best-response function of player k.

• An action profile (ai) is a Nash equilibrium if
• ak  Bk(a-k) for all player k.
• An action x of player k is a dominant action if
• x  Bk(a-k) for all action profile (a-k).

28
Exhaustive Method

• Begin with a game table.
• We will incrementally cross out outcomes
that are not Nash equilibria as follows:
• For each player k = 1..n
• For each profile (a-k) in A-k
• Compute v = maxxAk uk(a-k, x)
• Cross out all outcomes (a-k,x) such that uk(a-k, x) < v
• The remaining outcomes are Nash equilibria.

29
Example

Stand   Walk    Run

Float   62,65   38,74   34,32

Swim    68,38   55,52   31,36

Dive    33,37   32,30   22,28

30
Solution

Stand   Walk    Run

Float   62,65   38,74   34,32   74

Swim    68,38   55,52   31,36   52

Dive    33,37   32,30   22,28   37

68      55      34
31
Best-Response Table

Stand    Walk Run
Float 62,65    38,74 34,32
Swim 68,38     55,52 31,36
Dive 33,37     32,30 22,28

Stand Walk     Run              Stand Walk       Run
Float                   X         Float        X
Swim     X       X                Swim         X
Dive                              Dive    X
Row player’s best-response table Column player’s best-response table
32
Continuous Valued Strategies
• In many games, strategies can take on any
value in an interval of real numbers.
• Setting product prices
• Making investment
• Payoffs are usually given as functions of the
values of the strategies.
• Concepts of best responses, dominant
actions, dominated actions, Nash
equilibrium still apply

33
0          …           10                        A2
0     -6,24        …        -226,-56           30-(6+A2-2*A1)2,
A1
25-(1+A1-A2)2
…       …          …           …
Y1=30-(6+A2-2*A1)2
Y2=25-(1+A1-A2)2
10 -166,-96        …         14,24

Given a value of A2, what are the values of A1 that maximize Y1,
ie. what values are best responses of player 1 to the given value of A2.
Given a value of A1, what are the values of A2 that maximize Y2.

(a1,a2) is a Nash equilibrium if a1 is a best response to a2 and
a2 is a best response to a1.
34

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