ECON 101 – Problem Set 2

1. (Crime and Police)

. Here is a version of a model presented by a Nobel Price winner Gary

Becker, to analyze crime enforcement. The game has two players, Criminal (player 1) and Police (Player

2). Criminal chooses a level of crime

y0and Police chooses a level of law enforcement,x0.

The Police.s payo¤ is

??xc4??y2

x

, wherecrepresents cost of law enforcement, and the second term is

the negative e¤ect of crime of society. Criminal.s payo¤ is

py

1+

xy, wherepystands for the bene.t of

comitting crime, whereas

1

1+

xyis, roughly, the probability of evading capture.

a)

Write down the .rst order conditions that de.ne the best response functions and write down

the best response functions (Best response function for player

1speci.es what is his best response as a

function of action chosen by player

2, and vice versa). Graph the best response functions.

b)

Find the Nash Equilibrium of this game.

c)

What happens with the levels of law enforcement and crime (in Nash Equilibrium) when the cost

of law enforcement

cgoes up?

2.

Consider the Tragedy of Commons game from the class. Suppose that now, instead of farmers

choosing their actions simultaneously, player

1chooses his actionq1.rst, then player2observes this

choice and chooses

q2, after which the game ends (and payo¤s are as we had them in class).

a)

For anyq1, .nd the best response by player2.

b)

Find the Backward Induction solution of this game. What are the payo¤s to the players?

c)

Suppose now that before the sequential game that we just described player1can buy o¤ the

property rights to the .eld from player

2, and so become the sole owner of the .eld (in which case he

chooses

q1= 50and earns2500). More speci.cally, consider the extensive form game where in the .rst

round player

1makes an o¤ertbetween0and1500that is a multiple of100(and so has 16 actions

available,

t= 0;100; :::;1500). In the second round, player2can either accept or reject. If player 2

accepts

t, the game ends with payo¤s2500??tandt, respectively. If player2rejects the o¤er, players

play the extensive game as above (with player

1chosingq1in round3, followed by the choice ofq2by

player

2in round4). Find the Backward Induction solution of the game.

d) (extra – not graded!)

How would the answer change if we reversed the roles of player1and2

in the .rst two rounds of the game in c)?

3.

Back to the Tragedy of Commons from the class, in its simultaneous version.

a)

For each player, what are the dominated strategies in the original game? What is the action that

dominates them? (Hint: When looking for, say, dominated actions of player

1, it is very helpful to look

at the derivative of his pro.ts with respct to his action. Can this derivative be negative, no matter what

player

2does? What does it mean?)

b)

After you delete the dominated strategies in a), what are the dominated strategies in this “smaller”

game? (Formally: what strategies survive two rounds of deletion of dominated strategies)?

For your curiosity: the only strategies that survive the iterated deletion (if you continue) are the

Nash Equilibrium strategies

q1=q2= 331

3

. The proof is quite di¢ cult, however…

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