Game Theory
Professor Giacomo Bonanno
COOPERATIVE GAMES: the SHAPLEY VALUE
The description of a cooperative game is still in terms of a characteristic function which specifies for every group of players the total payoff that the members of S can obtain by
Definition of subgame. Start from a node that does not belong to an information set (more precisely, the
information set that contains that node consists of that node only). [In the example below, you can start
only at the root or at node y or at node x o
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
MIXED STRATEGIES

Given a reducedform strategic game N, (Si)i=1,n , (i)i=1,n , its mixed extension is
the reducedform game = N, (i) i=1,n, (Ei) i=1,
Department of Economics, University of California, Davis
Ecn 122  Game Theory  Professor Giacomo Bonanno TRUTHFUL REVELATION OF PREFERENCES UNDER THE PIVOTAL MECHANISM
A public project, say to build a road, is under consideration. The cost of the projec
Vickreys Second Price Auction
Consider first the case where there are 2 individuals and an object (e.g. a painting) to be
auctioned. Individual i values the object at $vi. Assume that each individual knows her own
valuation (but not necessarily the valuat
Ecn 122
Game Theory
Professor Giacomo Bonanno
The red paper / white paper on the back game
There are three children. The teacher announces that she is going to paste either a red or a white piece of paper on each childs back. Then the children are allowed
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
REPRESENTING KNOWLEDGE
Let us start with a single individual. Let W be the set of states or possible worlds. Each
state represents a (logically possible
Ecn 122  Game Theory  Professor Giacomo Bonanno
Department of Economics, University of California, Davis
STATIC GAMES OF INCOMPLETE INFORMATION
The distinction between complete and incomplete information is not at all the same as that between perfect an
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
HOW TO REVISE ONES BELIEFS: BAYES RULE
Suppose that all that matters is whether or not it rains and whether or not it is cold.
Then there are four possi
Department of Economics, University of California, Davis
Ecn 122  Game Theory  Professor Giacomo Bonanno
Expected utility theory
Games often involve lotteries, that is, probabilistic outcomes. For example, in an auction if two or more players submit the
ECONOMICS 122 GAME THEORY
Professor Giacomo Bonanno
~
HOMEWORK # 1 ANSWERS
1.
(a) The game is as follows:
Passenger 2
sit
stand
sit
2
2
3
0
stand
0
2
1
1
Passenger 1
(b) For each passenger sitting is a strictly dominant strategy.
(c) Both players will cho
ECONOMICS 122 GAME THEORY
Professor Giacomo Bonanno
~
HOMEWORK # 5 ANSWERS
1.
2.
The purestrategy Nash equilibria are (A,D) and (B,E). For player 1 C is strictly
dominated by A and for player 2 F is strictly dominated by E. Thus C and F must be
played wi
Game Theory
Professor Giacomo Bonanno
COOPERATIVE GAMES: the CORE
So far we have looked at noncooperative games, characterized by the fact that the individuals involved cannot sign binding agreements and therefore any suggested outcome has to be selfenf
ECONOMICS 122 GAME THEORY
Professor Giacomo Bonanno
~
HOMEWORK # 2 ANSWERS
(a) The extensive form is as follows:
Ann
keep
send
Barbara
reciprocate
$10
$10
keep
$25
$25
$0
$50
(b) The claim is not correct: in economics and game theory rationality is define
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
Seltens Chain Store Game
A chain store is a monopolist in an industry. It owns stores in m different towns. In each town the chain store makes $5m if le
ECONOMICS 122 GAME THEORY
Professor Giacomo Bonanno
~
HOMEWORK # 3 ANSWERS
1. (a) The extensive game is as follows:
Nature
H
1
T
0.8
0.2
T
1
T
2
H
H
T
H
H
T
2
H
T
3
1
H
1
0
2
1
0
0
3
0
1
1
T
2
0
0
1
(b) The corresponding normal form is as follows:
Player
ECONOMICS 122 GAME THEORY
Professor Giacomo Bonanno
~
HOMEWORK # 4 ANSWERS
(a) Let wwr represent the state where Ann has a white ball, Bob has a white ball and
Carla has a red ball, etc. Thus the set of possible states (given that there are only
two white
ECONOMICS 122 GAME THEORY
Professor Giacomo Bonanno
~
HOMEWORK # 6 ANSWERS
(a) The game is as follows:
NATURE
1q
q
R
2
0
R
1
1
L
2
0
L
2
E
F
E
F
4
2
0
4
4
4
0
2
(b) The strategic form is as follows:
Player 2
E
F
LL
2(2q)
0
2(1+q)
LR
2(1+q)
2q
2(1q)
4q
R
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
PRACTICE PROBLEMS for WEEK 10
T opics : (1 ) inco mplete info rma tio n ga mes
(2 ) w ea k s equentia l (or perfect Bay es ia n) equilibrium
VERY IMPORT
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
ANSWERS TO PRACTICE PROBLEMS for WEEK 10
1.
(a) In Harsanyis theory a strategy for a player specifies an action for each one of
his types. Albert has tw
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
ANSWERS TO PRACTICE PROBLEMS for WEEK 6
1.
The extensiveform representation of the simplified poker game is as follows (the
top number is Yvonnes net t
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
ANSWERS TO PRACTICE PROBLEMS for WEEK 9
1.
Since B is strictly dominated, it cannot be assigned positive probability at a Nash
equilibrium. Let p be the
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
ANSWERS TO PRACTICE PROBLEMS for WEEK 8
1.
The probabilities are as follows:
B
V
C
I
140 7
=
600 30
110 11
=
600 60
90
3
=
600 20
260 13
=
600 30
(a) Pr
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
ANSWERS TO PRACTICE PROBLEMS for WEEK 3
1.
(a) If they all report their true values, then wi = vi. Thus w1 + w2 + w3 + w4 = 40 50 + 29
+ 3 = 22 > C = 20
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
ANSWERS TO PRACTICE PROBLEMS for WEEK 4
1.
The Nash equilibria are (4,0), (3,1) and (4,2).
Bob
0
2.
2
0
0 ,5
0 , 10
0 ,5
1
0 , 10
0 ,5
5 , 0
2
0 ,5
5 ,
Department of Economics, University of California, Davis
Ecn 122 Game Theory Professor Giacomo Bonanno
ANSWERS TO PRACTICE PROBLEMS for WEEK 2
1.
In each of the six games (whether they are played by you or by the police) Not Pay strictly
dominates Pay. He
Department of Economics, University of California, Davis
Ecn 122  Game Theory  Professor Giacomo Bonanno
ANSWERS TO PRACTICE PROBLEMS for WEEK 5
1.
All the three extensive games have the following structure:
kill a
DN
Pay Not call police
free b kill
YOU
Ecn 122  Game Theory  Professor Giacomo Bonanno
Department of Economics, University of California, Davis
ANSWERS TO PRACTICE PROBLEMS for WEEK 1
1.
(a) Your ranking of the outcomes is: No kill Kill
Your ranking
The dognapper has the following ranking
Pa
122 Game Theory Professor Giacomo Bonanno
PRACTICE PROBLEMS for WEEK 6
Topic: IMPERFECTinformation games
VERY IMPORTANT: do not look at the answers until you have made a VERY serious effort to solve the problem. If
you turn to the answers to get clues or