1. problem set 4 from M. Osborne’s An Introduction to Game theory To view the problem set Click here →

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problem set 4from M. Osborne’s

An Introduction to Game theory

To view the problem set

Click here →

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Some (classical) examples of simultaneous games

CCooperate

D defect

C cooperate

3 , 3 0 , 6

D defect 6 , 0 1 , 1

Prisoners’ Dilemma

The ‘D strategy strictly dominates the C strategy

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Strategy s1 strictly dominates strategy s2 if for all strategies t of the other player

G1(s1,t) > G1(s2,t)

Strategy s1 weakly dominates strategy s2 if for all strategies t of the other player

G1(s1,t) ≥ G1(s2,t)and for some t

G1(s1,t) > G1(s2,t)

weakly

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1 , 5 2 , 3 7 , 4

3 , 3 4 , 7 5 , 2

X

X

Nash Equilibrium

Successive deletion of dominated strategies

example of strict dominance

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1 , 0 1 , 4 1 , 0 1 , 4

1 , 2 1 , 2 0 , 3 0 , 3

2 , 3 1 , 4 2 , 3 1 , 4

1 , 2 1 , 2 0 , 3 0 , 3

weak dominance

X

? ?

???an example

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Successive deletion of

dominated strategies

And Sub-game Perfectness

2

1

2

10 , 3 1 , 41 , 2

1 , 02 , 3

Examplerl

1

2

12

8

2

1

2

10 , 3 1 , 41 , 2

1 , 02 , 3

rl1

2

12

rr lr rl ll

rr

lr

rl

ll

rr lr rl ll

rr 1 , 0 1 , 4 1 , 0 1 , 4

lr 1 , 2 1 , 2 0 , 3 0 , 3

rl 2 , 3 1 , 4 2 , 3 1 , 4

ll 1 , 2 1 , 2 0 , 3 0 , 3

9

2

1

2

10 , 3 1 , 41 , 2

1 , 02 , 3

rl1

2

12

rr lr rl ll

rr 1 , 0 1 , 4 1 , 0 1 , 4

lr 1 , 2 1 , 2 0 , 3 0 , 3

rl 2 , 3 1 , 4 2 , 3 1 , 4

ll 1 , 2 1 , 2 0 , 3 0 , 3

Sub-game perfect equiibrium

( r,l ) ( l,l )

rr lr rl ll

rr 1 , 0 1 , 4 1 , 0 1 , 4

lr 1 , 2 1 , 2 0 , 3 0 , 3

rl 2 , 3 1 , 4 2 , 3 1 , 4

ll 1 , 2 1 , 2 0 , 3 0 , 3

10

rr lr rl ll

rr 1 , 0 1 , 4 1 , 0 1 , 4

lr 1 , 2 1 , 2 0 , 3 0 , 3

rl 2 , 3 1 , 4 2 , 3 1 , 4

ll 1 , 2 1 , 2 0 , 3 0 , 3

2

1

2

10 , 3 1 , 41 , 2

1 , 02 , 3

rl1

2

12

( x , l )

X

X

rr lr rl ll

rr 1 , 0 1 , 4 1 , 0 1 , 4

lr 1 , 2 1 , 2 0 , 3 0 , 3

rl 2 , 3 1 , 4 2 , 3 1 , 4

ll 1 , 2 1 , 2 0 , 3 0 , 3

( x , l )

X X

rr lr rl ll

rr 1 , 0 1 , 4 1 , 0 1 , 4

lr 1 , 2 1 , 2 0 , 3 0 , 3

rl 2 , 3 1 , 4 2 , 3 1 , 4

ll 1 , 2 1 , 2 0 , 3 0 , 3

delete ( x , r ) delete ( x , r )

weakly weakly dominating

X

rr lr rl ll

rr 1 , 0 1 , 4 1 , 0 1 , 4

lr 1 , 2 1 , 2 0 , 3 0 , 3

rl 2 , 3 1 , 4 2 , 3 1 , 4

ll 1 , 2 1 , 2 0 , 3 0 , 3

rr lr rl ll

rr 1 , 0 1 , 4 1 , 0 1 , 4

lr 1 , 2 1 , 2 0 , 3 0 , 3

rl 2 , 3 1 , 4 2 , 3 1 , 4

ll 1 , 2 1 , 2 0 , 3 0 , 3

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Another example of a simultaneous game

The Stag Hunt

Stag Hare

Stag 2 , 2 0 , 1

Hare 1 , 0 1 , 1

A generalization to n person game:

There are n types of stocks. Stock of type k yields payoff k if at least k individuals chose it, otherwise it yields 0.

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Another example of a simultaneous game

The Stag Hunt

Stag Hare

Stag 2 , 2 0 , 1

Hare 1 , 0 1 , 1

Stag Hare

Stag 2 , 2 0 , 1

Hare 1 , 0 1 , 1Equilibria

payoff dominant equilibrium

risk dominant equilibrium

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Stag Hare

Stag 2 , 2 0 , 1.99

Hare 1.99 , 0 1.99 ,1.99

Change of payoffs

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Yet another example of a simultaneous game

Battle of the sexes

Ballet Boxing

Ballet 2 , 1 0 , 0

Boxing 0 , 0 1 , 2

man

woman

Bach or Stravinsky (BOS)

Ballet Boxing

Ballet 2 , 1 0 , 0

Boxing 0 , 0 1 , 2

Equilibria

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Yet another example of a simultaneous game

Battle of the sexes

man

womanBallet Boxing

Ballet 2 , 1 0 , 0

Boxing 0 , 0 1 , 2

A generalization to a bargaining situation

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Nash Demand Game

Two players divide a Dollar.Each demands an amount ≥ 0.Each receives his demand if the total amount demanded is ≤ 1.Otherwise they both get 0.

Demand of player 1Dem

and

of

pla

yer

2

1

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Nash Demand Game

Demand of player 1Dem

and

of

pla

yer

2

1

equilibria

a continuum of equilibria

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last example of a simultaneous game

Matching Pennies

head tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1

no purepure strategies equilibrium exists

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Matching Pennies

head tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1


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Matching Pennies

head tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1


Mixed strategies

A player may choose

head with probability

and tails with probability 1-

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Matching Pennies Mixed strategies

head tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1

1- head

tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1

player 2 mixes:

if player 1 plays ‘head’ his payoff is the lottery:

1 - β β

1 -1

if the payoffs are in terms of his vN-M utility then his utility from the lottery is

β -1 + 1 - β 1 = 1 - 2β

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head tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1

1- head

tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1

player 2 mixes:

1 - 2β

Similarly, if player 1 plays ‘tails’ his payoff is …….

2β - 1

1 - β β

-1 1

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head tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1

1- head tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1

player 2 mixes:

1 - 2β

2β - 1

He prefers to play ‘head’ if:

1 - 2β > 2β - 10.5 > β

He prefers to play ‘tails’ if:

2β - 1 > 1 - 2ββ > 0.5

When β = 0.5 player 1 is indifferentbetween the two strategies

and any mix of the two

24player 1’s mix

player 2’s mix

α

(1-α , α)

β

1

1

head tails

Player 1 prefers to play ‘head’ if: 0.5 > ββ > 0.5Player 1 prefers to play ‘tails’ if:

Player 1 is indifferent when β = 0.5

Player 1’sBest Response

function


function ??

25player 1’s mix

player 2’s mix

α

β

1

1


function


function ??

When player 1 plays ‘head’ oftenPlayer 2 prefers to play ‘tails’


function

Nash equilibiumα = β= 0

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1/2head

1/2 tails

head 1 , -1 -1 , 1

tails -1 , 1 1 , -1

0

0

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Exercises from M. Osborne’s

An Introduction to Game Theory

EXERCISE 30.1 (Variants of the Stag Hunt) Consider variants of the n-hunter Stag Hunt in which only m hunters, with 2 ≤ m < n, need to pursue the stag in order to catch it. (Continue to assume that there is a single stag.) Assume that a captured stag is shared only by the hunters who catch it. Under each of the following assumptions on the hunters’ preferences, find the Nash equilibria of the strategic game that models the situation.

a. As before, each hunter prefers the fraction 1 / n of the stag to a hare

b. Each hunter prefers the fraction 1 / k of the stag to a hare, but prefers a hare to any smaller fraction of the stag, where k is an integer with m ≤ k ≤ n.

The following more difficult exercise enriches the hunters’ choices in the Stag Hunt. This extended game has been proposed as a model that captures Keynes’ basic insight about the possibility of multiple economic equilibria, some of which are undesirable (Bryant 1983, 1994).

Next exercise

28Next exercise

EXERCISE 31.1 (Extension of the Stag Hunt) Extend the n-hunter Stag Hunt by giving each hunter K (a positive integer) units of effort, which she can allocate between pursuing the stag and catching hares. Denote the effort hunter i devotes to pursuing the stag by ei , a nonnegative integer equal to at

most K. The chance that the stag is caught depends on the smallest of all the hunters’ efforts, denoted minj ej. (“A chain is as strong as its weakest link.”)

Hunter i’s payoff to the action profile (e1 . . ., en ) is 2minjej -ei . (She is better

off the more likely the stag is caught, and worse off the more effort she devotes to pursuing the stag, which means the catches fewer hares.) Is the action profile (e, . . . e), in which every hunter devotes the same effort to pursuing the stag, a Nash equilibrium for any value of e? (What is a player’s payoff to this profile? What is her payoff if she deviates to a lower or higher effort level?) Is any action profile in which not all the players’ effort levels are the same a Nash equilibrium? (Consider a player whose effort exceeds the minimum effort level of all players. What happens to her payoff if the reduces her effort level to the minimum?)

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2.7.5 Hawk-Dove

The Game in the next exercise captures a basic feature of animal conflict.

EXERCISE 31.2 (Hawk-Dove) Two animals are fighting over some prey. Each can be passive or aggressive. Each prefers to be aggressive if its opponent is passive, and passive if its opponent is aggressive; given its own stance, it prefers the outcome in which its opponent is passive to that in which its opponent is aggressive. Formulate this situation as a strategic game and find its Nash equilibria.

Next exercise

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EXERCISE 34.1 (Guessing two-thirds of the average) Each of three people announces an integer from 1 to K. If the three integers are different, the person whose integer is closest to 2/3 of the average of the three integers wins $1. If two or more integers are the same , $1 is split equally between the people whose integer is closest to 2/3 of the average integer. Is there any integer k such that the action profile (k,k,k), in which every person announces the same integer k, is a Nash equilibrium? (If k ≥ 2, what happens if a person announces a smaller number?) Is any other action profile a Nash equilibrium? (What is the payoff of a person whose number is the highest of the three? Can she increase this payoff by announcing a different number?)

Last excercise

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Game theory is used widely in political science, especially in the study of elections. The game in the following exercise explores citizens’ costly decisions to vote.

EXERCISE 34.2 (Voter participation) Two candidates , A and B, compete in an election. Of the n citizen, k support candidate A and m (= n - k) support candidate B. Each citizen decides whether to vote, at a cost, for the candidate she supports, or to abstain. A citizen who abstains receives the payoff of 2 if the candidate she supports wins, 1 if this candidate ties for first place , and 0 if this candidate loses. A citizen who votes receives the payoffs 2 - c, 1 - c, and -c in these three cases, where 0 < c < 1.

a. For k = m = 1, is the game the same (except for the names of the actions) as any considered so far in this chapter?

b. For k = m, find the set of Nash equilibria. (Is the action profile in which everyone votes a Nash equilibrium? Is there any Nash equilibrium in which one of the candidates wins by one vote? Is there any Nash equilibrium in which one of the candidates wins by two or more votes?)

c. What is the set of Nash equilibria for k < m?

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1. problem set 4 from M. Osborne’s An Introduction to Game theory To view the problem set Click here →