MATH 4321
Game Theory
Spring 2012
Assignment 5
1. Problems II.3.7.3, II.4.7.2, II.5.9.9, II5.9.10(d)
2. Solve the following matrix game using the method of linear
programming.
1 3
1 5
3. Given a matrix game suppose p1, p2 are optimal strategies for Player
MATH 310
Game Theory
Spring 2012
Assignment 4
1. Problems II.1.5.1, II.1.5.2, II.1.5.3
2. Problems II.2.6.2, II.2.6.4, II.2.6.5, II.2.6.6, II.2.6.7, II.2.6.8, II.2.6.10
3. Prove that for an mxn matrix game, any two saddle points have the
same value.
Missi
MATH 310
Game Theory
Spring 2012
Assignment 3
1. Reduce the Kuhn Tree of Exercise 2 in Assignment 2 to strategic form
and then find all PSEs.
2. Find the PPSE of the Votes by Veto game in Assignment 2.
3. Reduce the following Kuhn tree to strategic form.
MATH 4321
Game Theory
Spring 2012
Assignment 2
1. Problem II.5.9.1, II.5.9.4.
2. The Hidden pearl: There are two dark boxes. Player I hides a pearl
in one of them. Then Player II, not knowing which box contains the pearl,
peeks into one of them. If the pe
Recall that
Equilibrium Principle: BR to each other
Maximin Principle: Safety first
For Player I: Find p so that MinqpTAq is largest. p is
called the Safety Strategy or Optimal Strategy.
MinqpTAq is the lower value.
For Player II: Find q so that MaxppTA
Equilibrium Principle: BR to each other
Maximin Principle: Safety First
For Player I: Find p so that MinqpTAq is
largest. p is called the Safety Strategy or
Optimal Strategy.
MinqpTAq is the lower value.
For Player II: Find q so that MaxppTAq is
smallest.
Remark: In the above analysis, we used the basic
assumption of Common Knowledge.
A fact is common knowledge if everyone knows it, everyone
knows that everyone knows it, everyone knows that
everyone knows that everyone knows it,., and so on ad
infinitum.
,
Part II. Two-Person Zero-Sum Games
Example: Odd or Even
Players I and II simultaneously call out one of the numbers
one or two. Player Is name is Odd; he wins if the sum of the
numbers if odd. Player IIs name is Even; she wins if the sum
of the numbers is
Reduction of a Game in Extensive Form to Strategic
Form.
Pure strategy. A pure strategy is a players complete
plan for playing the game. It should cover every
contingency.
A pure strategy for a Player is a rule that tells him exactly
what move to make in
Goal: Give a complete mathematical description of the
game
game
Example:
1.Pick up bricks: A pile of 5 bricks has been stacked on the
Pick
ground. Two players take turn to pick up either one or two
bricks from the pile. The player who picks up the last br
Swastika Method to find all SEs for 2x2
games:
For 2x2 games, the players sets of strategies
are parameterized by
the unit interval.
We can make use of this to devise a graphical
method to find strategies that are best response
to each other.
For each mi
Overview: The analysis of two-person games
is necessarily more complex for general-sum
games than for zero-sum games. When the
sum of the payoffs is no longer zero (or
constant), maximizing ones own payoff is no
longer equivalent to minimizing the
opponen
Solving 2-Person 0-sum games by
Solving
sum
linear programming
linear
The basic problem of linear programming, determining the optimal value of
a linear function subject to linear constraints, arise in a wide variety of
situations.
In 1939the Russian math
9. The Sprague-Grundy Function.
Definition. The Sprague-Grundy function of a
graph, (X,F), is a function, g, defined on X and
taking non-negative integer values, such that
g(x) =mincfw_ n 0 : n g(y) for y F(x). (1)
In words, g(x) the smallest non-negative
Introduction
Game Theory can be called Interactive Decision Theory. It
studies the competition or cooperation between rational
and intelligent decision makers.
It has its origin in the entertaining games that people play,
such as chess, tic-tac-toe, bridg
Appendix 1: Utility Theory
Much of the theory presented is based on utility theory at a fundamental level. This
theory gives a justication for our assumptions (1) that the payo functions are numerical
valued and (2) that a randomized payo may be replaced
GAME THEORY
Thomas S. Ferguson
Part II. Two-Person Zero-Sum Games
1. The Strategic Form of a Game.
1.1 Strategic Form.
1.2 Example: Odd or Even.
1.3 Pure Strategies and Mixed Strategies.
1.4 The Minimax Theorem.
1.5 Exercises.
2. Matrix Games. Domination.