Lecture 11 Strategic Form Games - Purdue University · Lecture Outline 1 Overview of Game Theory 2...

Overview of Game TheoryStrategic Form Games and Dominant Strategies

Dominance Solvability

Lecture 11Strategic Form Games

Jitesh H. Panchal

ME 597: Decision Making for Engineering Systems Design

Design Engineering Lab @ Purdue (DELP)School of Mechanical Engineering

Purdue University, West Lafayette, INhttp://engineering.purdue.edu/delp

October 23, 2014

c©Jitesh H. Panchal Lecture 11 1 / 25

http://engineering.purdue.edu/delp



Lecture Outline

1 Overview of Game Theory

2 Strategic Form Games and Dominant Strategies

3 Dominance Solvability

Source: Dutta, P. K. (1999). Strategies and Games: Theory and Practice.Cambridge, MA, The MIT Press.




What is Game Theory?

Game Theory

Game theory is the study of strategic situations – situations in which an entiregroup of people is affected by the choices made by every individual withinthat group. It is about interdependence between decisions.

Can you think of some examples?




Let us play a game – Prisoner’s Dilemma

Consider two prisoners - A and B

B stays silent (cooperates) B betrays (defects)A stays silent (cooperates) Each serves 1 year A: 3 years, B: goes free

A betrays (defects) A: goes free, B: 3 years Each serves 2 years




Interdependence between Decisions

1 What will each individual guess about the others’ choices?2 What action will each person take?3 What is the outcome of these actions? Is this outcome good for the

group as a whole?4 Does it make any difference if the group interacts more than once?5 How do the answers change if each individual is unsure about the

characteristics of others in the group?




Formal Definition of a Game

1 Players: In any game, there is more than one decision maker; eachdecision maker is referred to as a “player”.

2 Interaction: What any one individual player does directly affects at leastone other player in the group.

3 Strategic: An individual player accounts for this interdependence indeciding what action to take.

4 Rational: While accounting for this interdependence, each playerchooses her best action.




Rules of the Game

To define a game, we need to specify four things:1 Who is playing – the group of players that strategically interacts2 What they are playing with – the alternative actions or choices, the

strategies, that each player has available3 When each player gets to play (in what order)4 How much they stand to gain (or lose) from the choices made in the

game.




Common Knowledge About the Rules

If you asked any player about who, what, when, and how much, they wouldgive the same answer.

This does not mean that all players have the same information when theymake choices, or are equally influential, or that all have the same choices. Itsimply means that everyone knows the rules.

Common knowledge goes a step further:

everyone knows the rules...

everyone knows that everyone knows the rules...

everyone knows that everyone knows that everyone knows the rules...

... ad infinitum




Forms of a game

1 Extensive form - generally used for sequential games2 Normal (strategic) form - generally used for simultaneous games




1. The Extensive Form of a Game

The extensive form is a pictorial representation of the rules of a game. Alsocalled a game tree.Nodes are decision nodes. Choices are branches.

Strategies: A pair of strategies (one for each player determines the way inwhich the game is actually played.)




Example of Extensive Form of a Game

Theater game: b = bus, c = car, s = subway, T = Ticket, N = No ticket.

Figure : 2.5 on page 24 (Dutta)





Game of Nim: Suppose there are two matches in one pile and a single matchin the other pile (2,1). The player who removes the last match wins the game.

Player 1

(2, 1)

Player 2

Player 2

Player 2

u

m

d

l

r

c

L

R

(1,1)

(0,1)

(2,0)

Player 1

(1,0)

(0,1)

(1,0)

(0,0)

(1,-1)

(1,-1)

(1,-1)

(-1,1)

(-1, 1)






Strategic Committee Voting: Voter thinks through what the other voters arelikely to do rather than voting simply according to the preferences.

Scenario: Suppose there are two competing bills (A, B) and three voters 1, 2,3. Possible outcomes are either bill passes or no bill passes (N).

Process: First, bill A is pitted against bill B. The winner is then pitted againstthe status quo (N).

The legislators have the following preferences:

1 A � N � B2 B � A � N3 N � A � B





Voter 1

A

B

A

B

A

B

Voter 2

Voter 3A

B

A

B

A

B

A

B

Voter 1A

N

A

N

A

N

Voter 2

Voter 3A

N

A

N

A

N

A

N

1,0,0

0, -1, 1





Information Sets and Strategies

Representing simultaneous moves within the extensive form.

Example: Player 2 is unable to distinguish between the two nodes (i.e.,whether Player 1 chose c or n).

Player 1

c

n

c

n

c

n

Player 2





2. The Normal Form of a Game

Example: Prisoner’s Dilemma

1 / 2 Confess Not ConfessConfess 0, 0 7,−2

Not Confess −2, 7 5, 5




Notation

Consider a set of players in a game labeled 1, 2, . . . ,N

i th player: A representative player

si : player i ’s strategies

s∗i : player i ’s specific strategy

s−i : a strategy choice of all players other than player i

s∗1 , s∗2 , . . . , s

∗N : a strategy vector (one strategy for each player)

πi(s∗1 , s∗2 , . . . , s

∗N): Player i ’s payoff for strategy vector s∗1 , s

∗2 , . . . , s

∗N




Example: Prisoner’s dilemma

Prisoner’s Dilemma

1 / 2 Confess Not ConfessConfess 0, 0 7,−2

Not Confess −2, 7 5, 5




Dominant Strategy

Dominant Strategy

Strategy s′i strongly dominates all other strategies of player i if the payoff to s′iis strictly greater than the payoff to any other strategy, regardless of whichstrategy is chosen by the other player(s). In other words,

πi(s′i , s−i) > πi(si , s−i)

for all si and all s−i




Weakly Dominant Strategy

Weakly Dominant Strategy

Strategy s′i (weakly) dominates another strategy, say s#i , if it does atleast as

well as s#i against every strategy of the other players, and against some it

does strictly better, i.e.,

πi(s′i , s−i) ≥ πi(s#i , s−i), for all s−i

πi(s′i , s−i) > πi(s#i , s−i), for some s−i

In this case, s#i is a dominated strategy. If s′i weakly dominates every other

candidate strategy si , then s′i is said to be a weakly dominant strategy.




Iterated Elimination of Dominated Strategies

Player 1 / Player 2 Left RightUp 1, 1 0, 1

Middle 0, 2 1, 0Down 0, -1 0, 0




Another example – Bertrand Competition

Firm 1 / Firm 2 High Medium LowHigh 6, 6 0, 10 0, 8

Middle 10, 0 5, 5 0, 8Low 8, 0 8, 0 4, 4




Iterated Elimination of Dominated Strategies:Advantages and Disadvantages

Advantage: simplicity

Disadvantages:1 Layers of rationality2 Order of elimination matters3 Non-unique outcomes

1 / 2 Left RightTop 0, 0 0, 1

Bottom 1, 0 0, 04 Nonexistence

1 / 2 Left Middle BadTop 1, -1 -1, 1 0, -2

Middle -1, 1 1, -1 0, -2Bad -2, 0 -2, 0 -2, -2




Summary

1 Overview of Game Theory

2 Strategic Form Games and Dominant Strategies

3 Dominance Solvability




References

1 Dutta, P. K. (1999). Strategies and Games: Theory and Practice.Cambridge, MA, The MIT Press.


THANK YOU!


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