M9302 Mathematical Models in Economics
Instructor: Georgi Burlakov
2.1.Dynamic Games of Complete and Perfect Information
Lecture 2
10.03.2011
OPVK_MU.tif

Fast Revision on Lecture 1
o
oStrategic Games of Complete Information:
o
oDescription
oNormal Form Representation
oSolution Concepts – IESDS vs. NE

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

Revision: Students’ Dilemma -2
(simultaneous-move solution)
oNash Equilibrium Solution:
o
YOU

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

Dynamic (sequential-move) games
o
oInformally, the games of this class could be described as follows:
o
oFirst, only one of the players chooses a move (action).
o
oThen, the other player(s) moves.
o
oFinally, based on the resulting combination of actions chosen in total, each player receives a
given payoff.

Example 1: Students’ Dilemma -2
(sequential version)
o
oStrategic behaviour of students taking a course:
o
oFirst, only YOU are forced to choose between studying HARD or taking it EASY.
o
oThen, the OTHERS observe what YOU have chosen and make their choice.
o
oFinally, both You and OTHERS do exam and get a GRADE.
o
Will the simultaneous-move prediction be defined?

o
oThe aim of the first lecture is to show:
o
o1. How to describe a dynamic game?
o
o2. How to solve the simplest class of dynamic games with complete and perfect information?
o
The dynamic (sequential-move) games

How to describe a dynamic game?
oThe extensive form representation of a game specifies:
o
1.Who are the PLAYERS.
o
o
o2.1. When each player has the MOVE.
o
o
o2.2. What each player KNOWS when she is on a move.
o
o
o2.3. What ACTIONS each player can take.
o
o
o3. What is the PAYOFF received by each player.
o
o

Example 1: Students’ Dilemma
(Sequential Version)
o
oExtensive Form Representation:
o
1.Reduce the players to 2 – YOU vs. OTHERS
o
o2.1. First YOU move, then – OTHERS.
o
o2.2. OTHERS know what YOU have chosen when
o
o they are on a move but YOU don’t.
o
o2.3. Both YOU and OTHERS choose an ACTION
o
o from the set     , for i = 1,…,n
o3.Payoffs:

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

Dynamic Games of Complete and Perfect Information
oThe simple class of dynamic games of complete and prefect information has the following general
description:
o
1.Player 1 chooses and action a1 from the feasible set A1.
2.
2.Player 2 OBSERVES a1 and then chooses an action a2 from the feasible set A2.
3.
3.Payoffs are u1(a1,a2) and u2(a1,a2).

Dynamic Games of Complete and Perfect Information
oStandard assumptions:
oPlayers move at different, sequential moments
o– it is DYNAMIC
oThe players’ payoff functions are common knowledge
o– it is COMPLETE INFORMATION
oAt each move of the game the player with the move knows the full history how the game was played
thus far
o– it is PERFECT INFORMATION
o
o
o

Example 1: Students’ Dilemma -2
(Sequential Version)
o
oStandard assumptions:
o
oStudents choose between HARD and EASY
o SEQUENTIALLY.
o
oGrading is announced in advance, so it is
o COMMON KNOWLEDGE to all the students.
o
oBefore making a choice in the second stage, OTHERS observe the choice of YOU in the first stage.
o
oSimplification assumptions:
o
oPerformance depends on CHOICE.
o
oEQUAL EFFICIENCY of studies.
o
o
o
o
o

Example 1: Students’ Dilemma – 2
(Sequential Version)
oGame Tree VS. Normal-Form
YOU
OTHERS
OTHERS
0
0
-3
-1
-2
-2
(HARD, HARD)
(HARD, EASY)
(EASY, HARD)
(EASY, EASY)
HARD
          -2,-2 (NE)
-2,-2
-1,-3
-1,-3
EASY
-3,-1
           0,0 (NE)
-3,-1
           0,0 (NE)
-1
-3
Easy
Hard
Easy
Hard
Easy
Hard

Backwards Induction
oSolve the game from the last to the first stage:
o
oSuppose a unique solution to the second stage payoff-maximization:
o
oThen assume a unique solution to the first stage payoff-maximization:
o
oCall  a backwards-induction outcome.

Example 1: Students’ Dilemma – 2
(Sequential Version)
(HARD, HARD)
(HARD, EASY)
(EASY, HARD)
(EASY, EASY)
HARD
          -2,-2 (NE)
-2,-2
-1,-3
-1,-3
EASY
-3,-1
           0,0 (NE)
-3,-1
           0,0 (NE)
YOU
OTHERS
OTHERS
-2
-2
-1
-3
0
0
0
0
Easy
Hard
Easy
Hard
Easy
Hard
0
0
-3
-1
-2
-2
(HARD, HARD)
(HARD, EASY)
(EASY, HARD)
(EASY, EASY)
HARD
          -2,-2 (NE)
-2,-2
-1,-3
-1,-3
EASY
-3,-1
         0,0 (SPNE)
-3,-1
        0,0 (NE)

Example 2: Students’ Dilemma -2
(with non-credible threat)
oStrategic behaviour of students taking a course:
o
oFirst, only YOU are forced to choose between studying HARD or taking it EASY.
o
oThen, the course instructor warns you:
oif YOU choose to study HARD in the first stage, all students get a WEAK PASS (C or 3)
oBut if YOU choose to take it EASY, OTHERS still have a choice and YOU are on a threat to FAIL (F
or 5)
o
Is instructor’s threat credible? Should YOU take it seriously?

Example 2: Students’ Dilemma – 2
(with non-credible threat)
o
oLeisure Rule: HARD study schedule devotes all the time (leisure = 0) to studying distinct from the
EASY one (leisure = 2).
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
2,2
-2,-2
0,0
At least one Hard
2,0
-5,-1
-3,-1
Hard
No Choice
0,2
-1,-1
-1,1

Example 2: Students’ Dilemma – 2
(with non-credible threat)
YOU
OTHERS
OTHERS
-1
1
0
0
0
0
Easy
Hard
Easy
Hard
0
0
-3
-1

Subgame Perfect Nash Equilibrium
o
oInformal Definition:
o
oThe only subgame-perfect Nash equilibrium is the backwards-induction outcome.
o
oThe backwards-induction outcome does not involve non-credible threats.

Summary
oDynamic (sequential-move) games represent strategic situations where one of the players moves
before the other(s) allowing them to observe her move before making a decision how to move
themselves.
oTo represent a dynamic game it is more suitable to use extensive form in which in addition to
players, their strategy spaces and payoffs, it is also shown when each player moves and what she
knows before moving.

Summary
oGraphically a dynamic game could be represented by the so called “game tree”.
othe number of the subgames is equal to the number of decision nodes in the tree minus 1.
oDistinct from the static games of complete information, here the strategy set of the second player
does not coincide with its set of feasible actions.
oStrategy in a dynamic game is a complete plan of action – it specifies a feasible action for each
contingency (other player’s preceding move) in which given player might be called to act.

Summary
oDynamic games of complete information are solved by backwards induction i.e. first the optimal
outcome in the last stage of the game is defined to reduce the possible moves in the previous
stages.
oBackwards induction outcome does not involve non-credible threats – it corresponds to the
subgame-perfect Nash equilibrium as a refinement of the pure-strategy NE concept.