Linköpings universitet
Ht-11
IEI/Nek
Göran Hägg
730A19. 733A55 Institutionell och spelteoretisk analys, 15 hp
Exercises and Problems in Game Theory
1) Consider the game in its normal form and draw it in its extensive form.
T1
T2
S1
10 , 0
5,2
S2
10 , 1
2,0
Mark all Nash equilibria.
2) Consider the game in its normal form and draw it in its extensive form.
left , left
left , right
right , left
right , right
Left
3,7
3,7
2,4
2,4
Right
6,1
8,2
6,1
8,2
Mark all Nash equilibria.
Is it always possible to translate a game in normal form into a game in extensive form?
3) Consider the games below in their extensive form. Draw the games in their normal form.
1
1
L
R
L
2
l
(5,1)
2
r
l
(0,0)
(0,0)
Mark all Nash equilibria.
R
r
(1,3)
l
(5,1)
2
r
(0,0)
l
(0,0)
r
(1,3)
4) Consider the games below and mark all possible Nash equilibria. Is there an obvious way to play
these games? Explain!
Game 1.
T1
T2
T3
S1
4,3
2,7
0,4
S2
5,5
5 , -1
-4 , -2
T1
T2
T3
S1
4 , 10
3,0
1,3
S2
0,0
2 , 10
10 , 3
Game 2.
Are ordinary people capable of solving strategic problems in real life in a rational way that take the
form of the games presented above?
How may we interpret possible Nash equilibria.
5) Consider the following game.
T1
T2
S1
10 , 0
5,2
S2
10 , 11
2,0
Mark all possible Nash equilibria.
Is there an obvious way to play this game?
Does the result change if the players are given the possibility to communicate before the game starts?
6) Write down and explain the formal definition of a Nash equilibrium.
What components does a strategic game with perfect recall and perfect information consist of?
2
7) Consider the following game.
T1
T2
T3
T4
S1
0,0
0,0
100 , 100
0,0
S2
0,0
0,0
0,0
99 , 99
S3
100 , 100
0,0
0,0
0,0
S4
0,0
100 , 100
0,0
0,0
Mark all Nash equilibria.
Does there exist any reason to believe that S2,T4 will be a solution to the game. Explain!
What type of game is this?
8) Consider the following game and identify all Nash equilibria.
r
U
1
D
(0,0)
2
(2,2)
l
(3,1)
Is there an obvious way to play this game? Explain!
9) Explain and compare the solution concepts of (i) Nash equilibrium, (ii) Subgame perfection, (iii)
Backward induction, and (iv) Dominance solvable. Can a Nash equilibrium be identified (in a game
with perfect information) by means of Backward induction but still not be Subgame perfect?
10) Consider the game below.
1
2
A
D
(98,98)
1
2
A
D
A
D
A
(100,100)
D
(97,100) (99,99) (98,101)
How do we solve this game? Is there any Nash equilibria? Is there any subgame perfect solution? If
there is a solution, will rational players play it?
3
11) What is a proper subgame?
12) Consider the game below. How many proper subgames do there exist? Draw them.
1
C
D
2
2
E
F
(2,1)
(3,0)
G
H
(0,2)
(1,3)
13) Consider the game below.
R
U
1
D
2
r
(1.5,1.5)
L
(1,1)
(0,1)
1
l
R
(3,3)
L
(4,0)
Identify possible Nash equilibria.
Identify the proper subgames.
Does there exists a subgame perfect equilibrium?
14) Consider the game below.
T1
T2
S1
-100 , -100
2,2
S2
2,2
-100 , -100
Identify all possible Nash equilibria.
4
Is there any obvious way to play this game?
What kind of game is it? What kind of social interaction does it represent?
Can you give an explanation why one equilibrium will be established?
15) Consider the following game.
T1
T2
S1
3,1
1,3
S2
0,5
4,2
Does there exists any Nash equilibria?
Calculate the solution.
Is this solution a plausible equilibrium in a social interaction that resembles this game? Explain!
16) Consider the following one shot game (the game is only played once) where player 1 chooses between the strategies S1 and S2, while player 2 choose between the strategies T1 and T2:
T1
T2
S1
4,4
3,5
S2
5,3
0,0
a) Identify all Nash equilibria and the expected payoff (expected utility) of each player.
b) Can the players improve their payoffs by cheap talking to each other before the game or by
preplay negotiation? What is the problem?
c) Present the game in its extensive form.
d) Assume that the game is played repeatedly forever and that the two players are patient—i.e.,
they discount the payoffs and their discount factors are close to 1. What does the folk theorem
say about this game? Try to make a thorough analysis by applying the folk theorem.
e) Is there any pair of strategies that seems to be more likely than others when the game is repeated? Why?
f) Now, assume that the discount factor of player 1 is low, i.e., close to zero, while it is still high
for player 2. Does this change the possible outcome of the game? Try to make an analysis.
5
17) Consider the following game.
T1
T2
S1
3,3
1,4
S2
4,1
2,2
What kind of game is this?
Does the game have one or several equilibria if it is repeated 100 times by the same players, and the
players know that it will end after 100 times of play?
Does the solution you suggest agree with your experience of how people plays games like this in real
life? Give one example.
What happens if the game is repeated indefinitely, in regard to possible equilibria?
Is S2,T2 a possible equilibrium if the game is repeated indefinitely?
Is S1,T1 a possible equilibrium if the game is repeated indefinitely?
18) Consider the following game. There exist 11 letters: A, B, C, D, E, F, G, H, I, J, and K. Each letter
has a secret value measured in Swedish crowns, between 1 and 100 Skr. The value is generated by a
random-machine. The game is played between Player 1 and Player 2. Independently of each other the
players has to pick a sample of the letters and write them down. Player 1 has to write down letter D
and Player 2 has to pick letter K. For each letter put down on paper by one player, but not taken by the
other player, the former player receives the amount of money of which the letter represents. If a letter
is put down by both players, they have to pay a fine which is the double amount of the value of which
the letter represents. If the players manage to divide all the letters between them, they will receive the
triple amount of money of which the letters represents.
Does there exists any Nash equilibrium in this game? How many?
Does there exists any obvious way to play? Explain!
How would you play if you know that Paul Nystedt is Player 1 and you are Player 2?
19) Consider the following game, which we will call the ”truth game.” There are two players, called 1
and 2, and a game-master. The game-master has a coin that is bent in such a way that, flipped randomly, the coin will come up “heads” 80% of the time. (The bias of this coin is known to both players.)
The game-master flips this coin, and the outcome of the coin flip is shown to player 1. Player 1 then
makes an announcement to player 2 about the results of the coin flip; player 1 is allowed to say either
“heads” or “tail” (and nothing else). Player 2, having heard what player 1 says but not having seen the
results of the coin flip then must guess what the result of the coin flip was – either “heads” or “tails.”
That ends the game. Payoffs are made as follows. For player 2 things are quite simple; player 2 gets $1
if he guess matches the actual results of the coin flip, and he gets $0 otherwise. For player 1 things are
more complex. She gets $2 if player 2’s guess is that the coin came up “heads”, and $0 if player 2
guesses “tails”, regardless of how the coin came up. In addition to this, player 1 gets $1 (more) if what
she (player 1) says to player 2 matches the results of the coin flip, while she gets $0 more if her message
to player 2 is different from the result of the coin flip.
6
Draw an extensive form representation of this game.
Convert this extensive form representation to a normal form representation.
20) Consider the following game.
Left
Right
Upp
1,1
5,0
Down
0,5
4,4
Identify the Nash equilibrium if the game is played only once.
Demonstrate that the Pareto-efficient solution of Down,Right can be a subgame perfect Nash equilibrium if the game is infinitely repeated and both players adopt the the following trigger strategy:
Play Down,Right initially or if this has always been played in the past. Otherwise play Up,Left.
How does the necessary condition on the value of the discount factor change if the players adopt the
following alternative punishment strategy for the infinitely repeated game?
Play Down,Right initially or if the outcomes Down,Right or Up,Left occurred in the previous period. Otherwise play Up,Left for one period.
Explain the result!
How many Nash equilibria do there exist in this game if it is repeated infinitely?
7
21) Consider the following game.
T1
T2
S1
8,8
2 , 10
S2
10 , 2
4,4
The game is repeated indefinitely. The return from the game (utility) is ui t  , where the index i represents player 1 or player 2 ( i  1,2 ). The argument t stand for the stage of the game ( t  0,1,2,3,..., 
). The first stage of the repeated game we assign as t  0 . For example, if the payoff in stage 155 for
player 1 is 8, we express it as u1 155  8 . Moreover, assume that both players discount future payoffs
by means of the discount factor   0.8 . The sum of all future discounted payoffs for player i can
a t  ui t  . Both players tries to maximize the sum of all future discounted
payoffs – maximize the payoff in the supergame.
then be written as


t 0
If both players play the strategy “tit for tat”, what will the payoff for the supergame then be?
If both players always choose the strategies S2 and T2, what will the payoff for the supergame be?
For which values of the discount factor  will player 1 play the strategy T2 even if player 2 is expected
to play “tit for tat”?
Does “tit for tat” represent a Nash equilibrium?
Analyse the game by means of the Folk Theorem.
22) Consider the position of Indiana Jones in the climax of the movie Indiana Jones and the last crusade.
Indiana Jones, his father and the Nazis have all converged at the site of the Holy Grail. The two Joneses refuse to help the Nazis reach the last step. So the Nazis shoot Indiana’s dad. Only the healing power of the Holy Grail can save the senior Dr. Jones from his mortal wound. Suitably motivated, Indiana
leads the way to the Holy Grail. But there is one final challenge. He must choose between literally
scores of chalices, only one of which is the cup of Christ. While the right cup brings eternal life, the
wrong choice is fatal. The Nazi leader impatiently chooses a beautiful golden chalice, drinks the holy
water, and dies the sudden death that follows from a wrong choice. Indiana picks a wooden chalice,
the cup of a carpenter. Exclaiming: “There’s only one way to find out”! He dips the chalice into the
font and drinks what he hopes is the cup of life. Upon discovering that he has chosen wisely, Indiana
brings the cup to his father and the water heals the mortal wound.
Analyse the story above as a strategic game. Do the parties choose strategies rationally? Could they
have done better?
23) Roulette is a strategic game. Analyse the following strategic situation.
At the end of his academic studies, Barry celebrated at one of Cambridge University’s May Balls. Part
of the festivities included a casino. Everyone was given £20 worth of chips, and the person who had
amassed the greatest fortune by evening’s end would win a free ticket to next year’s ball. When it came
time for the last spin of the roulette wheel, by a happy coincidence, Barry led with £700 worth of
chips, and the next closest was a young Englishwomen with £300. The rest of the group had been ef8
fectively cleaned out. Just before the last bets were to be placed, the women offered to split next year’s
ball ticket. Barry refused.
To better understand the next strategic move, we take a brief detour to the rules of roulette. The betting in roulette is based on where a ball will land when the spinning wheel stops. There are typically
numbers 0 through 36 on the wheel. When the ball lands on zero, the house wins. The safest bet in
roulette is to bet on even or odd (denoted by Black or Red). These bets pay an even money – a on
pound bet returns two pound – while the chance of winning is only 18/37.
The Englishwomen bet her entire stake on the chance that the ball would land on a multiple three.
This bet pays two to one (so her £300 bet would return £900 if she won) but has only a 12/37 chance
of winning.
Did Barry make the right move by refusing to split the next year’s ball ticket?
Do Barry have an optimal response to the strategy of the Englishwomen? Explain!
Did the Englishwomen play an optimal strategy?
24) How can relatively small armies of occupying powers or tyrants control very large populations for
long periods?
Analyse the problem by means of game theory and give an explanation.
25) In the United States many homeowners own guns for self-defence. In Sweden almost no one
owns a gun. Cultural differences provide one explanation. The possibility of strategic moves provides
another.
In both countries, a majority of homeowners prefer to live in an unarmed society. But they are willing
to buy a gun if they have reason to fear that criminals will be armed. Many criminals prefer to carry a
gun as one of the tools of their trade.
The table below suggests a possible ranking of outcomes.
Criminals
No Guns
Guns
No Guns
4,3
1,4
Guns
3,1
2,2
Homeowners
Which solutions are possible if the game is played simultaneously? Explain!
Discuss the problems with guns in society by means of the game and suggest possible ways of establishing No Guns, No Guns as equilibrium.
26) King Lear was worried about how his children would treat him in his old age. Much to his regret,
he discovered that children do not always deliver what they promise. In addition to love and respect,
children are also motivated by the possibility of an inheritance. Here we look at how a strategic use of
inheritance can manipulate children to visit their parents.
9
Imagine that parents want their children each week to each visit once and phone twice. To give their
children the right incentives, they threaten to disinherit any child who fails to meet this quota. The estate will be evenly divided among all the children who meet this quota. The children recognize that
their parents are unwilling to disinherit all of them. As a result, they get together and agree to cut back
the number of visits, potentially down to zero.
The parents call you in and ask for some help in revising their will. Where there is a will, there is a way
to make it work. But how? You are not allowed to disinherit all of the children.
27) An auctioneer invites bids for 100 Skr. Bidding proceeds in steps of 5 Skr. The highest bidder gets
100 Skr., but both the highest and the second highest bidders pay their bids to the auctioneer.
Analyse the game and suggest possible Nash equilibria.
Does it exist situations in real life that resembles this game. Give an example.
28) The U.S. government had a major problem trying to motivate several million teenagers to register
for the military draft. Large-scale civil disobedience would make it impossible to punish everyone who
violates the law. Still, the government had a big advantage: it sets the rules.
To see the advantage of moving first, imaging that the government is only allowed to punish one person who fails to register. How can they use this single threat to induce everyone to register?
29) Developing a new commercial airplane is a gigantic gamble. The cost of designing a new engine
alone can reach two billions of dollars (1990). It is no exaggeration to say that building a new and better plane requires “betting the company”. No wonder governments get involved, each trying to make a
larger market for its domestic firm.
Here we look at the market for 150-passenger medium-range jets: the Boeing 727 and the Airbus 320.
Boeing developed the 727 first. Did it make sense for Airbus to enter the market?
The primary market for these aircraft was in the United States and in the Europe. We assume each of
these markets is worth $900 million to a monopoly firm. Were the two firms to compete head-on, total
profit fall from $900 to $600 million, divided evenly between the two firms. Although profits fall,
competition results in cheaper planes and lower airfares, so consumers benefit. These benefits to consumers are worth $700 million in each market.
Airbus Industries estimates that it will cost $1 billion to develop the Airbus 320. If they go ahead without any government assistance, they can expect to make a profit of $300 million in each of the markets,
Americana and EU. The total of $600 million is not enough to cover the development costs. The EU
governments cannot offer direct assistance in the form of subsidies because their budget is already
committed to subsidizing farmers.
You are called to Brussels and asked for advice on whether the EU should assist Airbus by giving it a
protected market, that is, requiring European airlines to buy the Airbus 320 over the Boeing 727. What
do you suggest? How do you expect the United States government to respond?
30) Do the exercises and problems in Carmichael.
31) Betrakta spelet nedan som spelas mellan A och B.
B
10
Tuff
A
Svag
Tuff
- 6, - 6
12, 4
Svag
4, 12
2, 2
a) Genomför en spelteoretisk analys och redovisa samtliga möjliga Nash jämvikter.
Spelförutsättningarna är följande: Spelet spelas bara 1 gång mot en slumpmässigt vald
motspelare som du inte har någon som helst information om. Spelarna känner till spelet och
har spelat det tidigare mot slumpmässigt valda motståndare. Payoffen är von NeumanMorgernstern nyttor.
b) Översätt spelet i uppgift a) till extensiv form och genomför en spelteoretisk analys. Har
någonting förändrats?
c) I det samhälle som du lever i finns det en fördelning mellan Tuffa och Svaga typer enligt
följande: Pr(Tuff )  0,4 och Pr( Svag )  0,6 . Tuffa typer väljer strategin Tuff och svaga
typer väljer strategin Svag. Du kan inte se på utsidan om motspelaren är en Tuff typ eller en
Svag typ. Spelet spelas fortfarande bara 1 gång mot slumpmässigt valda motståndare.
Genomför en spelteoretisk analys över hur du skall spela spelet givet att du är riskaversiv samt
givet att du är en risk lover.
d) Vi bygger vidare på uppgift c). I det samhälle du lever i finns det motståndare som har
glasögon och som inte har glasögon. Du kan observera om motspelaren har glasögon eller ej.
Enligt din subjektiva uppfattning är de med glasögon generellt tuffare typer enligt följande
fördelning:
Pr(Glasögon Tuff )  0,75 ;
Pr(Glasögon Svag)  0,25 ;
Pr( Ej _ Glasögon Tuff )  0,25 ;
Pr( Ej _ Glasögon Svag)  0,75
Genomför en spelteoretisk analys över hur du skall spela spelet givet att du möter motståndare
som har glasögon och som ej har glasögon.
e) Anta nu istället att vi spelar spelet ovan upprepat utan slut mellan två motståndare. Payoffen är
som tidigare von Neuman – Morgenstern nyttor. Båda spelarna har diskonteringsfaktor
  0,8 . Genomför en spelteoretisk analys och identifiera möjliga Nash jämvikter. (Observera
att vi inte utgår ifrån sannolikhetsfördelningarna i uppgift c) och d) utan att detta är ett rent
upprepat spel mellan två spelare.)
f) Anta som i uppgift e) att vi spelar spelet ovan upprepat utan slut mellan två motståndare.
Payoffen är som tidigare von Neuman – Morgenstern nyttor. Båda spelarna har
diskonteringsfaktor   0,8 . Enda skillnaden mot uppgift e) är att spelet är sekventiellt och att
spelare A tar beslut före spelare B. Spelare B kan observera spelare A:s val av drag. Genomför
en spelteoretisk analys och identifiera möjliga Nash jämvikter. (Hint: konstruera ett spelträd
som speglar spelreglerna i delspelet.)
g) Anta att spelet i uppgift f) ovan ofta spelas mellan olika spelare i ett samhälle. Motståndarna
som du möter dras slumpmässigt ur en urna och du spelare bara en gång mot varje
motståndare. Du tar alltid rollen av spelare B medan dina motspelare tar rollen av spelare A,
11
det vill säga, du väljer om du skall spela Tuff eller Svag efter att motspelaren valt att vara Tuff
eller Svag, och detta kan du observera. När ni spelat sprids det information om hur du spelat
och om utgången av spelet. Informationen är allmänt känd. Vad har du för spelteoretisk
möjlighet att se till att du i denna situation får så hög genomsnittlig payoff som möjligt genom
att spela spelet upprepat mot nya motståndare som väljer drag före dig? Förklara!
32) Anta att de olika formerna av spelet i fråga 3 ovan illustrerar/representerar en konfliktsituation
mellan två grupper av samhällsmedborgare där spelare A och spelare B tillhör två urskiljbara grupper
som spelar mot varandra. Det kan till exempel vara en konfliktsituation mellan kvinnor och män. Det
kan också representera en konfliktsituation mellan gamla och unga, eller rika och fattiga. En grupp
kanske utmärker sig med stora muskler medan den andra har små muskler, osv. Spelet kan
representera många olika situationer. Hur som helst, en Nash jämvikt i vissa former av spelet kan
uppfattas som att det illustrerar djupt orättvisa och diskriminerande förhållanden som kan fortgå i ett
samhälle. Genom en etablerad Nash jämvikt kan man tänka sig att den ena gruppen kan komma att
betraktas som stark medan den andra betraktas som svag, osv.
Diskutera utifrån spelen i fråga 3 och din analys genom hela fråga 3 möjligheten att etablera Nash
jämvikter som kan uppfattas som ”rättvisa” och ej diskriminerande. Med andra ord, möjligheter att
förebygga eller motarbeta orättvisa och diskriminering. Under vilka förutsättningar kan vi uppnå
”rättvisa” eller ej diskriminerande lösningar. Vad kan vi göra i det fall det inte finns möjlighet till
”rättvisa” eller ej diskriminerande Nash lösningar.
Observera att essäsvaret på denna fråga skall baseras på en spelteoretisk analys/diskussion.
12