EC302 Political Economics

Game theory refresher

Prof. Ronny Razin

October 2013

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Rationality and Game Theory in Political Economics?

Political situations often involve a small number of players.

Strategic thinking - Solutions concepts in which a player doesn’t know
what the equilibrium is, have to guess what the other players are
doing and that way decide on the course of action. The tool -
assumption of Rationality.
Information - Through actions of others deducting their information,
because it might be relevant to my decision.
Example: Polls, Party names...

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Important to remember

A game is a mathematical interpretation of real-life scenarios. A tool

for economic insights.
Some cases the insights are quite interesting and enlightening. In
other games, the results might be counter intuitive and ridiculous.
We will:
1 Learn how to describe political economic problems using game theory.
2 Learn about the di¤erent solution concepts of game theory.
3 Learn to make sensible decisions in using game theory and choosing the
right solution concepts.

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Rationality
Description

"Rationality"- a model we use to describe behaviour.

1 A decision maker (DM)- i.

2 A set of actions - A.
3 A preference relation of the set of outcomes, represented by the utility
function ui : A ! <.

Note: what is implicitly assumed in (3)?

Completeness
Transitivity, and...
Continuity
Note: at this stage we discuss only ordinal utilities – no lotteries.

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Strategic (Normal-form) Game
Description:

1 Group of players N = f1, ..., n g

2 For each player i 2 N a set of actions - Ai , (A = ∏ Ai ).
i 2N
3 For each player i a preference relation of the set of outcomes,
represented by the utility function ui : A ! <.

Players choose their action simultaneously.

Note: at this stage we discuss only ordinal utilities – no lotteries.
But will assume cardinal utities when we disscuss mixed strategies.

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Strategic (Normal-form) Game
Rationality in a game:

Each player has beliefs about the other players’actions (a i ).

Chooses an action ai 2 Ai , that maximizes u (ai , a i ) given her beliefs.

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Strategic (Normal-form) Game
Example – Prisoner’s dilemma

C D
C 1, 1 10, 0
D 0, 10 9, 9
If players choose their action independently, what will happen in this
game? Why?
Rational behavior of each player will result in an ine¢ cient outcome
(not Pareto).
Relevant examples from the real world?

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Strategic (Normal-form) Game
De…nition - Dominant Strategy

De…nition
Action ai 2 Ai strictly dominates ai0 2 Ai if for all a i 2A i

u (ai , a i ) > u (ai0 , a i )

“if a rational player has a strictly dominant action, he will choose it.”

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Strategic (Normal-form) Game
Example: Dominance - Hoteling’s game

Two parties are in an election - "Right" and "Left"

Political view point are represented by a number between 1 and -1.

Voters are uniformly distributed between all of the positions.

The Game:
Each party chooses a position on the line:
The Right party in the area AR = [0,1];
The Left party in the area AL = [-1,0].
Then each voter votes to the party closes to his ideal point.
Where will each party position itself?

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Strategic (Normal-form) Game
Dominance - Iterated elimination of strictly dominated actions

Back to the prisoner’s dilemma.

Add the action - Attempt escape (E).
Attempting escape always fails and the punishment is 20 years in jail.
The police has no longer use for a witness. Hence if the other player
(not the escaped one) confesses to the police, get 9 years. If keeps
silent - 1 year.
C D E
C 1, 1 10, 0 1, 20
The new matrix for the game:
D 0, 10 9, 9 9, 20
E 20, 1 20, 9 20, 20
D is not strictly dominating C anymore, but it is obvious that E is
strictly dominated.
After eliminating E, we can eliminate C.
"If a rational player has a strategy that is dominated by others, she
will never choose it"
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Strategic (Normal-form) Game
Iterated elimination of strictly dominated strategies

Lemma
There is no importance for the order in which strictly dominated strategies
are eliminated.

Proof (set 1)

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 W X Y Z
A 5, 5 6, 4 3, 3 7, 1
B 4, 1 0, 3 0, 4 5, 12
C 0, 1 0, 0 4, 6 12, 0
D 0, 10 10, 0 2, 8 6, 10

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Strategic (Normal-form) Game
Weakly dominated strategies

De…nition
Action ai 2 Ai weakly dominates ai0 2 Ai if (i) for all a i 2A i

u ( ai , a i ) u (ai0 , a i )

and (ii) there exists an a0 i 2 A i such that

u (ai , a0 i ) > u (ai0 , a0 i )

Iterated elimination: order matters.

L C R
U 1, 1 1, 1 1, 1
M 1, 0 0, 1 1, 1
D 1, 1 1, 0 0, 0

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Strategic (Normal-form) Game
The Byzantine Generals (or the email game)

2 generals on mountains want to conjure a valley.

If 1 attacks – loses; if 2 attack – victory. Need to strike together.
A fog is in the valley. There is a need to coordinate the attack –
sending a messenger.
There is a slight chance that the messenger won’t arrive, hence
sending a con…rmation and so on.
What will happen?

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Strategic (Normal-form) Game
De…nition - Common Knowledge

De…nition
The fact F is a common knowledge if "Every player knows F"; "Every
player knows that every other player knows F"; Every player knows that
every player knows that every player knows F" and so on.

Theorem
If, between players, there is a common knowledge of the game and the fact
that all are rational, then the result of the game must be among the
results that survive an iterated elimination of strictly dominated strategies.

Assumptions needed in order to use dominating strategies: "Every

player is rational and knows her own payo¤s".

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Strategic Form Games
Mixed Strategies: defenition and notation

De…nition
A mixed strategy of Player i, αi , is a lottery on her pure strategies,
α i 2 ∆ ( Ai ) .

αi , A function αi : Ai ! [0, 1] such that ∑ αi (ai ) = 1.

a i 2A i
The mixed strategies set contains the pure strategies as well.
Notations: pro…le of mixed strategies, α = (α1 , ..., αn ).
When the players are playing mixed strategies α, the probability that
n
action pro…le a will be played is: α(a) = ∏ α i ( ai )
i =1
Independence assumption!

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Strategic Form Games
Mixed Strategies: defenition and notation

What are the preferences of the players on the expanded outcome set?
Use expected utility:
" #
n
ui ( α ) = ∑ α(a)ui (a) = ∑ ∏ αi (ai ) ui (a1 , ..., an )
a 2A a 2A i =1

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Strategic Form Games
Mixed Strategies: example

L R
U 10, 0 0, 2
M 4, 2 4, 0
D 0, 0 10, 2
Suppose α1 = ( 14 , 12 , 14 ) (short for α1 (U ) = 14 , α1 (M ) = 21 ,
α1 (D ) = 14 )
Suppose α2 = ( 13 , 23 ) meaning α2 (L) = 13 , α2 (R ) = 23 .

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Strategic Form Games
Mixed Strategies: example

L R L R
1 1
U 10, 0 0, 2 U 12 6
)
M 4, 2 4, 0 M 6 31
1
1 1
D 0, 0 10, 2 D 12 6
Suppose α1 = ( 14 , 12 , 14 ) (short for α1 (U ) = 14 , α1 (M ) = 21 ,
α1 (D ) = 14 )
Suppose α2 = ( 13 , 23 ) meaning α2 (L) = 13 , α2 (R ) = 23 .
What is player 1 utility?
1 1 1 1
u1 = 10 + 4+ 4+ 10 = 4.5
12 6 3 6
And for Player 2?
1 1 1
u2 = 2+ 2+ 2=1
6 6 6

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Strategic Form Games
Nash Equilibrium: best response

De…nition
Action ai 2 Ai is a best response to a i 2A i if

ai 2 BR (a i ) = arg max u (ai , a i )

a i 2A i

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Strategic Form Games
Nash Equilibrium: best response

X Y W Z
A 5, 2 3, 6 0, 2 2, 2
Example, B 7, 6 7, 7 3, 2 2, 0
C 3, 2 2, 3 3, 1 2, 0
D 2, 1 9, 2 2, 0 2, 1
BR1 (X ) = fB g, BR1 (Y ) = fD g, BR1 (W ) = fB, C g,
BR1 (Z ) = fA, B, C , D g
BR2 (A) = BR2 (B ) = BR3 (C ) = BR4 (D ) = fY g

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Strategic Form Games
Nash equilibrium

De…nition
The action pro…le (a1 , . . . , an ) is a Nash equilibrium (in pure strategies) If
for each player i, ai 2 BR (a i ).

In the example
BR1 (X ) = fB g, BR1 (Y ) = fD g, BR1 (W ) = fB, C g,
BR1 (Z ) = fA, B, C , D g
BR2 (A) = BR2 (B ) = BR3 (C ) = BR4 (D ) = fY g
(D,Y) is the unique Nash eq.

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Strategic Form Games
Nash equilibrium: examples

C D B S H T
C 3, 3 1, 4 B 2, 1 0, 0 H 1, 1 1, 1
D 4, 1 1, 1 S 0, 0 1, 2 T 1, 1 1, 1

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Strategic Form Games
Nash equilibrium: interpretations

More problematic compared to previous concepts. Requires that:

1 Each player has the correct belief about other players’actions.
2 Each player is rational in a sense that she chooses the optimal action
give her belief.
The …rst requirement is very problematic, especially in games with
more then one equilibrium. In which cases is it reasonable?
1 Norm – everybody knows what side to drive on the road.
2 Preliminary talks – reaching to an agreement that is self-enforcing.
3 Stable solution (over time) - The game is played many times. The
players choose the action that is optimal based on the past (disregard
any strategic considerations at the moment).
Nash eq. is a necessary condition for stability (but not su¢ cient and
not necessarily can be easily converge to).
Similar to norm.

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Strategic Form Games
Bertrand competition

Company i …xes price Pi. Manufacturing cost 0.

A consumer buys one unit of good from the cheapest company.
Invalid graph:

p2(p1)

p1(p2)

The intersection is at (0, 0) – unique Nash.

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Strategic Form Games
Bertrand competition

But this is not the convenient (or correct) way to …nd Nash eq.
Another way to describe Nash eq.:
A strategy pro…le in which no player can bene…t from deviating.
(Coincide with interpretation of stable Nash – no one has an interest to
break it).
Bertrand – unique Nash (0, 0), since from every other (Pi , Pj ) there is
a pro…table deviation.
Important – Nash eq. is immune to deviation by an individual player,
not groups.

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Strategic Form Games
Mixed strategies

Extend the de…nition of BR to include mixed strategies:

BRi (α i ) = arg max ui (αi , α i )

Nash equilibrium:

(α1 , ..., αn ) such that for all i 2 N, αi 2 BRi (α i )

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Strategic Form Games
Mixed strategies: Example

Second example: Battle of the sexes

q 1-q

p 2,1 0,0

1-p 0,0 1,2

When will player 2 use a mixed strategy?

Only when the two pure strategies give the same payo¤:
p + (1 p )0 = 0p + 2(1 p) ,
3p = 2,
2
p =
3
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Strategic Form Games
Mixed strategies: Example

Second example: Battle of the sexes

q 1-q

p 2,1 0,0

1-p 0,0 1,2

When will player 1 use a mixed strategy?

Only when the two pure strategies give the same payo¤:
2q + 0(1 q) = 0q + 1(1 q) ,
3q = 1,
1
q =
3
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Strategic Form Games
Mixed strategies

In general:
In order to …nd equilibrium in mixed strategies, player i mixed strategy
needs to be such that player j will be indi¤erent between his pure
strategies.
Is this convincing?

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Strategic Form Games
Mixed strategies: Graphical representation

1
8
q> 3 <p=1
1
q= 3 ) BR1 (q ) = p 2 [0, 1]
1 :
q< 3 p=0
2
8
p> 3 < q=1
2
p= 3 ) BR2 (p ) = q 2 [0, 1]
2 :
p< 3 q=0

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Strategic Form Games
Mixed strategies: Graphical representation

q
1

BR1(q)

3/1
BR2(p)
0 p
0
3/2 1

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Strategic Form Games
Mixed strategies: Battle of the sexes continued

Equilibria:
1 (p, q ) = (0, 0) (ballet, ballet) - pure, payo¤ (2,1)

2 (p, q ) = (1, 1) (football, football) - pure, , payo¤ (1,2)

3 (p, q ) = ( 2 , 1 ) – mixed
3 3
What are the payo¤s in the mixed eq.?
1 2
3 3
2
3 2, 1 0, 0 )
1
3 0, 0 1, 2
1 2
3 3
2 2 4
For player 1: 3 9 2 9 0 ) u1 ( 23 , 13 ) = 6
9 = 2
3
1 1 2
3 9 0 9 1
1 2
3 3
2 2 4
For player 2: 1
3 9 0 ) u2 ( 23 , 13 ) = 69 = 23
9
1 1 2
0
3 9 2 9
Both players are better o¤ with one of the pure strategy equilibria.
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Strategic Form Games
Mixed strategies: Battle of the sexes continued

Is the mixed eq. here convincing?

Why should each player play to keep the other player indi¤erent?
In considering a dynamic process of convergence to eq., then it is more
likely that both will converge to a pure one (since the mixed eq. is not
stable).
If there is prior consent – the pure eq. is better
Is mixed eq. never convincing?

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Strategic Form Games
The case of Kitty Genovese

(New York Time article - March 27th, 1964):

http://www.garysturt.free-
online.co.uk/The%20case%20of%20Kitty%20Genovese.htm
March 64’. A woman is murdered in Queens. 38 neighbors hear but
no one calls 911.
Model:
n identical neighbors.
Cost (individual) for calling: 1.
Utility (per individual) if one calls the police: x > 1
Nash eq. in pure strategies: exactly one calls. Improbable without
coordination.

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Strategic Form Games
The case of Kitty Genovese

Look for mixed eq: Every neighbor calls with probability p.

Every neighbor needs to be indi¤erent between calling and not calling:
Payo¤ if calls: x 1.
Payo¤ if doesn’t call: x (1 (1 p )n 1)
Indi¤erence:

x 1 = x (1 (1 p )n 1
),
1 1
p = 1 ( )n 1
x
Hence the probability that someone will call 911 is
1 1 1 n
1 (1 p )n = 1 (( ) n 1 )n = 1 ( )n 1
x x

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Strategic Form Games
The case of Kitty Genovese

In equilibrium: the probability that someone will call 911 is

1 n
1 (1 p )n = 1 ( ) n 1
x
For n = 1 this equal 1.
For a large n the expression gets smaller and goes to 1

1-1/x

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Strategic Form Games
Nash equilibrium: Existence

Existence of Nash equilibrium (very brie‡y)

Does every game have a Nash eq.?
Pure – not necessarily, as we seen in the odd/even game.
Mixed?
Theorem: Every game with a …nite number of actions has at least one
Nash eq. (in mixed)

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