Math-Comp 553: Algorithmic Game Theory.
Assignment 4: Solutions
1. Zero-Sum Games. Consider a general two-player game. The Player 1 (the row
player) has payo u1 (r, c) and Player 2 (the column player) has payo u2 (r, c),
if row r and column c are selected
Math-Comp 553: Algorithmic Game Theory.
Practice Exam : Solutions
1. Nash Equilibria.
(a) What is a Nash equilibria in a 2-player game?
A pair of strategies that are mutual best responses.
(b) Consider a game with the payo matrices A for the row player an
Math-Comp 553: Algorithmic Game Theory.
Practice Exam 1
Instructions. This exam is 3 hours long and contains 9 questions. Answer
as many question as you can. (Full marks can be obtained for correct answers
to seven questions.) You may quote any result/the
Math-Comp 553: Algorithmic Game Theory.
Practice Exam 2: Solutions
1. Mimimax Theorem.
(a) State the minimax theorem.
Take any nite, 2-player, zero-sum game. Both players receive a payo equal to
their maxmin and minmax values.
(b) Consider a zero-sum game
Math-Comp 553: Algorithmic Game Theory.
Assignment 6: Solutions
1. Matching Markets.
(a) At these prices, the corresponding utilities are
a(3) b(1) c(4) d(2)
I
3
6
2
7
7
4
5
6
II
III
6
5
4
5
6
10
3
6
IV
The preference sets for each buyer are then: SI = cf
Math-Comp 553: Algorithmic Game Theory.
Assignment 5 Solutions
1. Combinatorial Auctions: Single-Minded Bidders.
Order the bidders such that v1 v2 vn and then apply the greedily
algorithm. We accept the highest bid (that is, allocate S1 to v1 ) and then
r
Math-Comp 553: Algorithmic Game Theory.
Assignment 1: Solutions
1. Nash Equilibria
(a) Assume P1 plays T with probability p and B with probability 1 p; assume
that P2 plays L with probability q and R with probability 1 q . In a mixed
NE with 0 < p < 1, P1
Math-Comp 553: Algorithmic Game Theory.
Assignment 2: Solutions
1. VCG Mechanisms.
(a) The social welface with choice S is w(S ) = v1 (S ) + v2 (S ) + v3 (S ) cS .
Looking at the table, we have w(N ) = (2 7 + 5) 0 = 4; w(L) =
(0 + 9 + 6) 4 = 11; w(H ) = (
Math-Comp 553: Algorithmic Game Theory.
Assignment 3: Solutions
1. NP-Completeness. What we want to do is modify the game used to prove that
FAIR-NASH is NP-Complete in such a way that the new game has exactly one
Nash Equilibrium if and only if the under
Math-Comp 553: Algorithmic Game Theory.
Practice Exam 2
Instructions. This exam is 3 hours long and contains 9 questions. Answer
as many question as you can. (Full marks can be obtained for correct answers
to seven questions.)
1. Mimimax Theorem.
(a) Stat