Homework Assignment 2
Due on Nov 25
Problems can be discussed in groups. If you discuss the solutions with fellow students
please list the students’ names on your solutions, and do not submit verbatim copies of
others’ solutions.
Please do not look at the solution manual of the OR book.
Worth 19 points.
Problem 1
(B&O)
Consider the following zero-sum game on the unit square
u(x1 , x2 ) = x31 − 3x1 x2 + x32 ,
where x1 is the action of Player 1, the maximizer, and x2 is the action of Player 2, the minimizer.
a) Determine the pure or mixed strategy saddle point equilibria.
(3p)
(Hint: Analyze the properties of the utility function. What is the best response of the players?)
Problem 2
Proper equilibria
Consider the following 3 person strategic game (Table 1). As usual, player 1 is the row player and player
2 is the column player. Player 3 plays the two tables.
L
H
L
H
1,1,1 0,0,0
0,0,0 0,0,0
(L)
L
H
L
H
0,0,0 0,0,0
0,0,0 1,1,0
(H)
Table 1: Payoff tables for Problem 2
a) Show that both (L,L,L) and (H,H,L) are proper equilibria. (Find a sequence of -proper strategies
for these equilibria that converges.)
(2p)
(For hints see E.Kalai and D.Samet, Persistent equilibria in strategic games, International Journal of
Game Theory 13(3), pp. 129-144, 1984.)
Problem 3
(Exercise 48.1 from OR, Course in Game Theory)
Consider the following three player game given in normal form (Table 2). Player 1 chooses between T
and B, Player 2 chooses between L and R, Player 3 chooses between X, Y and Z.
a) Show that the pure strategy equilibrium payoffs are (1,0,0), (0,1,0) and (0,0,0).
(1p)
b) Show that there is a correlated equilibrium in which player 3 chooses Y and players 1 and 2 choose
(T,L) and (B,R) with equal probabilities.
(2p)
c) What would be the equilibrium if player 3 had the same information as players 1 and 2?
1
(1p)
Homework 2
T
B
FEP3301 Computational Game Theory P2/2015
L
R
0,0,3 0,0,0
1,0,0 0,0,0
(X)
T
B
L
R
2,2,2 0,0,0
0,0,0 2,2,2
(Y)
T
B
L
R
0,0,0 0,0,0
0,1,0 0,0,3
(Z)
Table 2: Payoff table for Problem 3
Problem 4
(Exercise 28.1 from OR, Course in Game Theory)
Each of two bowls contains a number of pieces of paper. Each paper contains a number from a finite subset
S of the interval [0,1]. The numbers in the two bowls are identically distributed and their distribution is
described by the distribution function F. Consider the following two player strategic game. Each player
knows the elements of the set S but does not know the distribution F. Each player picks a piece of
paper from a different bowl (without seeing the number on it). After looking at the number, each player
has to decide simultaneously whether to exchange the piece of paper with the other player. If both of
them decide for the exchange then they exchange the pieces of papers. After the decision (and eventual
exchange) each player receives a prize worth the number written on the piece of paper she has. The goal
of both players is to maximize their prize.
a) Model this situation as a Bayesian game and show that in any Nash equilibrium the highest number
that a player is willing to exchange is the smallest possible number independent of the distribution
F.
(3p)
Problem 5
Fictitious play
Consider the following zero-sum game.

3
 2
1
3
5
0
1
6
7

4
3 
0
Use the following procedure to compute the strategies of the two players.
• Start with an arbitary pure strategy for each player at step 1.
• At each step, compute the empirical mixed strategy for each player based on her actions up until
the previous step. (E.g., if Player 1 has played action i ni times during the first n steps then xi is
estimated to be ni /n.
• At each step, find the best response for each player based on
Pthe other player’s emprirically observed
mixed strategy. (E.g., for Player 2 choose j to minimize i aij xi .
a) Does the sequence of the empirical strategies converge to a saddle point? Include a copy of the
computer program that you use to simulate the process and a figure that shows some points of the
sample path of empirical mixed strategies to support your answer.
(3p)
Problem 6
Potential game
Consider the following problem. There are N nodes and there is a pool of O objects. Object o has
weight wio for node i. Each node can store one object from the pool of objects. The nodes are located
on a graph G = (I, E), E is such that G is connected, and the maximum node degree in the graph is
K < |O|. The set of neighbors of player i ∈ I is denoted by Ni .
Node i can use the object stored locally and the objects stored at the neighboring nodes. The goal
of every node is to maximize the sum of the weights of distinct objects it can use (i.e., the weight of the
object stored locally and at some neighbors counts only once).
a) Model this problem as a strategic game!
(1p)
b) Show the existence of pure strategy Nash equilibria.
(1p)
2
Homework 2
FEP3301 Computational Game Theory P2/2015
c) Show that every sequence of improvement steps, in which at each step there is only one node that
changes its stored object, is finite (that is, the game has the FIP).
(2p)
(Hint: Observe that the utility function of a player can be simplified compared to its nave form to a utility
function whose maximization is equivalent (consider the gain of storing an object given the neighbors’
actions). Use the simplified utility function to construct an exact, weighted, ordinal or generalized ordinal
potential function, whichever is possible.)
3