Game Theory and Applications
Problem set 1
1. For bimatrix games defined by matrices




3 0
0
−1 2 4
• A =  2 5 −1 , B =  −2 2 1 ,
3 1
2
4 0 1




1 1 2
2 4 3
• A =  0 0 1 , B =  0 3 4 ,
−1 3 0
2 2 1
(a) Find all the Nash equilibria.
(b) Apply the iterated dominance algorithm. Can you find some crucial difference between these two
games resulting in the fact that only in one of them an equilibrium can be found by the algorithm?
2. Find a 3 × 3 bimatrix game with a single Nash equilibrium, which is deleted from the strategy sets of the
players in the iterated dominance algorithm.
(a) Show that if only strictly dominated strategies are deleted in the iterated dominance algorithm, no
Nash equilibria can be deleted from the game.
(b) Show that any Nash equilibrium of a game obtained after the iterated dominance algorithm is a Nash
equilibrium in the initial game.
3. Find all the Nash equilibria in a 3-player game where Player 1 chooses a row, Player 2 chooses a column
and Player 3 chooses a layer of 3-dimensional payoff matrices
2 1
1 1
A(·, ·, 1) =
A(·, ·, 2) =
1 0
0 3
B(·, ·, 1) =
C(·, ·, 1) =
0
1
2
1
0
0
2
2
4
2
3
0
3
3
0
0
B(·, ·, 2) =
C(·, ·, 2) =
Are there any dominated strategies in this game?




1 2 0
1 2 3
4. Consider a bimatrix game with payoff matrices A =  2 2 3 , B =  2 1 2 . Show that it has
3 0 1
0 3 1
two Nash equilibria. Further, show that the output of the iterated dominance algorithm is always one of
them, but the order of deleting dominated strategies in the algorithm decides upon which one.
5. K players want to buy some object. Each of them has a private valuation of it vk (denominated in units
of some currency), so if he buys it for pk units, his utility is vk − pk . The decision about the buyer and
the price he has to pay can be reached in different ways. In first-price auction players write their offers
on pieces of paper and hand them over to the auctioneer in closed envelopes. Then the auctioneer opens
the envelopes and sells the object to the highest bidder for the price he has written. If there are two (or
more) players with the same highest bid, the good is awarded with equal probability to each one of them
(and only one pays for what he has received). In second-price auction the procedure is the same, but the
price paid by the winner equals the proposition of the second-highest bidder.
(a) Show that each player writes his true valuation vk in the equilibrium of the second-price auction.
Moreover, writing the truth is the dominating strategy for each player.
(b) Show that telling the truth about own valuation is not an equilibrium strategy in the first-price auction.
(c) How do you think, in reality auctions of which type give higher revenue to the auctioneer?
6. Suppose two players are repeatedly playing a strategic form game with closed strategy sets X, Y ⊂ R
and jointly continuous utility functions u1 , u2 . Let (xn , y n )∞
n=0 be a sequence of strategic movements (as
defined on the lecture) by the players starting in some arbitrary initial strategies x0 ∈ X and y 0 ∈ Y .
Show that if (xn , y n ) −→ (x∗ , y ∗ ) then x∗ and y ∗ form a Nash equilibrium in the game.
7. (Partnership Game) Suppose Greg (Player 1) and Mary (Player 2) are business partners. Each of the
partners has to determine the amount of effort he or she will put into the business, which is denoted by
xi , i = 1, 2 and may be any nonnegative real number. The cost of effort xi for Player i is cxi , where c > 0
is a fixed unit cost equal for both players. The success of business depends on the amount of effort put
1 α2
in it by the players – the profit denoted by r(x1 , x2 ) = xα
1 x2 , where α1 , α2 ∈ (0, 1) are fixed constants
known by both players, is shared equally between the two partners. Each player’s utility is given by the
difference between the player’s share of the profit and the cost of his/her effort put into the business.
(a) Describe this situation as a normal-form game.
(b) Find all the Nash equilibria in this game.
(c) Show that whenever the game has a unique Nash equilibrium, the sequence of strategic movements of
the players converges to this equilibrium from any initial point (x01 , x02 ) such that x0i 6= 0 for i = 1, 2.