M9302 Mathematical Models in Economics
Instructor: Georgi Burlakov
0.Game Theory – Brief Introduction
Lecture 1
24.02.2011
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What is Game Theory?
oWe do not live in vacuum.
o
oWhether we like it or not, all of us are strategists.
o
oST is art but its foundations consist of some simple basic principles.
o
oThe science of strategic thinking is called Game Theory.
o
o
o

Where is Game Theory coming from?
oGame Theory was created by
oVon Neumann and Morgenstern (1944)
oin their classic book
oThe Theory of Games and Economic Behavior
o
oTwo distinct approaches to the theory of games:
o1. Strategic/Non-cooperative Approach
o2. Coalition/Cooperative Approach
o
o

oThe key contributions of John Nash:
o
o1. The notion of Nash equilibrium
o
o2. Arguments for determining the two-person bargaining problems
o
oOther significant names:
oN-Nash, A-Aumann, S-Shapley&Selten, H-
oHarsanyi
Where is Game Theory coming from?

M9302 Mathematical Models in Economics
Instructor: Georgi Burlakov
1.1.Static Games of Complete Information
Lecture 1
26.02.2010
OPVK_MU.tif

The static (simultaneous-move) games
o
oInformally, the games of this class could be described as follows:
o
oFirst, players simultaneously choose a move (action).
o
oThen, based on the resulting combination of actions chosen in total, each player receives a given
payoff.

Example: Students’ Dilemma
o
oStrategic behaviour of students taking a course:
o
oFirst, each of you is forced to choose between studying HARD or taking it EASY.
o
oThen, you do your exam and get a GRADE.
o

Static Games of Complete Information
oStandard assumptions:
oPlayers move (take an action or make a choice) simultaneously at a moment
o– it is STATIC
oEach player knows what her payoff and the payoff of the other players will be at any combination
of chosen actions
o– it is COMPLETE INFORMATION
o
o

Example: Students’ Dilemma
o
oStandard assumptions:
o
oStudents choose between HARD and EASY
o SIMULTANEOUSLY.
o
oGrading policy is announced in advance, so it is
o known by all the students.
o
oSimplification assumptions:
o
oPerformance depends on CHOICE.
o
oEQUAL EFFICIENCY of studies.
o
o
o
o
o

o
oGame theory answers two standard questions:
o
o1. How to describe a type of a game?
o
o2. How to solve the resulting game-theoretic problem?
o
The static (simultaneous-move) games

How to describe a game?
oThe normal form representation of a game contains the following elements:
o
1.PLAYERS – generally of number n
2.
2.STRATEGIES –  , for i = 1,…,n
3.
3.PAYOFFS –                                ,for i = 1,…,n
o
oWe denote the game of n-players by
o G =
o
o
o
o

Example: Students’ Dilemma
o
oNormal Form Representation:
o
1.Reduce the players to 2 – YOU vs. OTHERS
2.
2.Single choice symmetric strategies
§ , for i = 1,…,n
§
3. Payoff function:
o

Example: Students’ Dilemma
o
oGrading Policy:
o
othe students over the average have a STRONG PASS (Grade A, or 1),
othe ones with average performance get a WEAK PASS (Grade C, or 3) and
owho is under the average
o FAIL (Grade F, or 5).

Example: Students’ Dilemma
o
oLeisure Rule: HARD study schedule devotes twice more time (leisure = 1) to studying than the EASY
one (leisure = 2).
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
2
3
-1
At least one Hard
Hard
At least one Easy
All Hard
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
Hard
At least one Easy
1
1
0
All Hard
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
2
5
-3
Hard
At least one Easy
All Hard
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
Hard
At least one Easy
All Hard
1
3
-2
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
Hard
At least one Easy
All Hard

Example: Students’ Dilemma
o
oBi-matrix of payoffs:
o
o

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2
YOU
OTHERS

How to solve the GT problem?
o
o
o
o
oSubgame-Perfect Nash Equilibrium (SPNE)
o
oBayesian Nash Equilibrium (BNE)
o
oPerfect Bayesian Equilibrium (PBNE)
o
in static games of complete information
in dynamic games of complete information
in static games of incomplete information
in dynamic games of incomplete information
Solution Concepts:
oStrategic Dominance
oNash Equilibrium (NE)

Strategic Dominance
oDefinition of a strictly dominated strategy:
o
oConsider the normal-form game G =
o
oFeasible strategy      is strictly dominated by strategy
o
o if i’s payoff from playing     is strictly less
o than i’s payoff from playing    :
o
o
o for each feasible combination
o that can be constructed from the other players’
o strategy spaces     .

Strategic Dominance
o
oSolution Principle: Rational players do not play strictly dominated strategies.
o
oThe solution process is called “iterated elimination of strictly dominated strategies”.

Example: Students’ Dilemma
oSolution by iterated elimination of strictly dominated strategies:

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2
YOU
OTHERS

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2
Easy is strictly dominated by Hard for YOU.
Easy is strictly dominated by Hard for OTHERS.
After elimination a single strategy combination remains:
{HARD; HARD}

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2

Weaknesses of IESDS
o
oEach step of elimination requires a further assumption about what the players know about each
other’s rationality
oThe process often produces a very imprecise predictions about the play of the game

Example: Students’ Dilemma -2
o
oLeisure Rule: HARD study schedule devotes all their time (leisure = 0) to studying.
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
2
3
-1
At least one Hard
2
5
-3
Hard
At least one Easy
1
All Hard
3
0
1
-1
0
3
-3
1
0
1
-2

Example: Students’ Dilemma -2
oSolution by iterated elimination of strictly dominated strategies:
o

Easy
Hard
Easy
-1,-1
-3,-1
Hard
-1,-3
-3,-3

Easy
Hard
Easy
-1,-1
-3,-1
Hard
-1,-3
-3,-3
YOU

Easy
Hard
Easy
-1,-1
-3,-1
Hard
-1,-3
-3,-3

Easy
Hard
Easy
-1,-1
-3,-1
Hard
-1,-3
-3,-3

Easy
Hard
Easy
-1,-1
-3,-1
Hard
-1,-3
-3,-3

Easy
Hard
Easy
-1,-1
-3,-1
Hard
-1,-3
-3,-3

Easy
Hard
Easy
-1,-1
-3,-1
Hard
-1,-3
-3,-3
No single strategy could be eliminated:
{EASY/HARD; EASY/HARD}
OTHERS

Nash Equilibrium
oDefinition (NE): In the n-player normal form game
o
oG =
o
othe strategies       are a Nash equilibrium if,
ofor each player i,
o    is (at least tied for) player i’s best response to the strategies
ospecified for the n-1 other players:
o
o
ofor every feasible strategy    in     ; that is,     solves

Relation between Strategic Dominance and Nash Equilibrium
o
oIf a single solution is derived through iterated elimination of strictly dominated strategies it
is also a unique NE.
o
oThe players’ strategies in a Nash equilibrium always survive iterated elimination of strictly
dominated strategies.

o
oGrading Policy:
o
othe students over the average have a STRONG PASS (Grade A, or 1),
othe ones with average performance get a PASS (Grade B, or 2) and
owho is under the average
o FAIL (Grade F, or 5).
Example: Students’ Dilemma - 2

Example: Students’Dilemma - 2
o
oLeisure Rule: HARD study schedule devotes all their time (leisure = 0) to studying than the EASY
one (leisure = 2).
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
2
At least one Hard
2
5
-3
Hard
At least one Easy
0
1
-1
All Hard
0
2
0
2
-2
3
-1
3
-3

Example: Students’ Dilemma -2
oSolution by iterated elimination of strictly dominated strategies:
o

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2
YOU

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2
No single strategy could be eliminated:
{EASY/HARD; EASY/HARD}
OTHERS

Example: Students’ Dilemma -2
oNash Equilibrium Solution:
o

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2
YOU

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2
Two Nash Equilibria:
{EASY/EASY; HARD/HARD}
OTHERS

Example: Students’ Dilemma - 2
o
oSome useful policy implications:
o
oHarsh grading of the mediocre behavior would motivate the rational students to study hard.
o
oExtremely time-consuming studies discourage rational students and make them hesitant between
taking it easy and studying hard.
o

Summary
oThe simplest class of games is the class of Static Games of Complete Information.
oBy ‘static’ it is meant that players choose their strategies simultaneously without observing each
other’s choices.
o‘Complete information’ implies that the payoffs of each combination of strategies available are
known to all the players.
oStatic games of complete information are usually represented in normal form consisting of
bi-matrix of player’s payoffs.
o

Summary
oA strategy is strictly dominated if it yields lower payoff than another strategy available to a
player irrespective of the strategic choice of the rest of the players.
oThe weakest solution concept in game theory is the iterated elimination of strictly dominated
strategies. It requires too strong assumptions for player’s rationality and often gives imprecise
predictions.
oNash Equilibrium is a stronger solution concept that produces much tighter predictions in a very
broad class of games.