1

Introduction

APEC 8205: Applied Game Theory

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Objectives

• Distinguishing Characteristics of a Game• Common Elements of a Game• Distinction Between Cooperative & Noncooperative Game Theory• Describing Extensive Form Games• Strategy in Extensive Form Games• Normal Form Games• Classification of Games

3

Distinguishing Characteristics of a Game

• An Individual’s Actions Affect the Welfare of Others• Others’ Actions Affect the Welfare of an Individual• Individuals Realize the Effect of Their Actions on Others• Implications:

– Must Consider What Others Will Do

– Must Consider What Others Think You Will Do

This is different from assuming the actions of others are exogenously given.

4

Common Elements of a Game

• For all games we must answer 4 basic questions:– Who are the players?

– Who can do what when?

– Who knows what when?

– How are players rewarded based on what they do?

The answers to these questions are the rules of the game.

These rules are assumed to be common knowledge!

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Example: Matching Pennies

• Who are the players?– Mason & Spencer

• Who can do what when?– Both can choose either Heads or Tails.

– Both make their choice at the same time.

• Who knows what when?– Neither player knows the choice of the other before making their own.

• How are players rewarded based on what they do?– Spencer pays Mason $1 if both choose Heads or both choose Tails.

– Mason pays Spencer $1 if one player chooses Heads & the other Tails.

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Example: Unfair Matching Pennies

• Who are the players?– Mason & Spencer

• Who can do what when?– Both can choose either Heads or Tails.

– Spencer makes his choice first.

• Who knows what when?– Mason gets to see Spencer’s choice before making his own choice.

• How are players rewarded based on what they do?– Spencer pays Mason $1 if both choose Heads or both choose Tails.

– Mason pays Spencer $1 if one player chooses Heads & the other Tails.

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Distinction Between Cooperative & Noncooperative Game Theory

• Both Start by Describing the Rules of the Game• For cooperative game theory, predictions for play are deduced

based on set of axioms (e.g. Pareto optimality, fairness, or equity).• For noncooperative game theory, predictions for play are deduced

based on the assumption that players act individualistically to maximize their own benefit.

This class focuses on noncooperative games!

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Describing Extensive Form Games

• Elements of a Game Tree:– V: Set of Nodes

– A: Set of branches

– r: Root Node

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Example Game Tree

v1

v3

v2

v4

v5

v6

v7

V = {v1, v2, v3, v4, v5, v6, v7}

A = {(v1, v2), (v1, v3), (v2, v4), (v2, v5), (v3, v6), (v3, v7)}

r = {v1}

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Some Definitions & Restrictions

• Path: A sequence of nodes connected by branches.– Restriction: Any two nodes are connected by a single path.

• Root: The beginning of the tree.• Incoming Branches: Branches on a path to the Root.

– Restriction: The Root has no incoming branches.

• Outgoing branches: Branches not on a path to the Root.• Terminal Nodes/Leaves: Nodes with no outgoing branches.

– Represent the end to the tree.

• Non-Terminal/Decision Nodes: Nodes with outgoing branches.– Represent points in the tree where decisions can be made.

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Examples

v1

v3

v2

v4

v5

v6

v7

Example Path = v1, v2, v4

For v2, (v1, v2) is incoming, while (v2, v4) and (v2, v5) are outgoing.

Terminal Nodes = {v4, v5, v6 , v7}

Decision Nodes = {v1, v2, v3}

Not a Path = v1, v2, v6

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Not a Tree

v1

v3

v2

v4

v5

v6

Multiple paths from v1 to v6:

v1, v3, v6

v1, v2, v5, v3, v6

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In Addition to the Tree, We Must

• describe the players: N = {0,1,2,…,n} where player 0 is a special player referred to as chance or Nature.

• partition the non-terminal nodes in the tree to the players: P0, P1,…, Pn.

• assign actions or choices to each branch of the tree. • assign probability distributions over outgoing branches for each

node in P0.

• partition the nodes into k(i) information sets for each player i: U1i, U2

i, …, Uk(i)

i.

• describe an n-dimensional vector, g(t) = (g1(t), g2(t),…, gn(t)), of payoffs for each terminal node t.

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Unfair Matching Pennies Example

v1

v3

v2

v4

v5

v6

v7

Spencer (S)

Mason (M)

Mason (M)

Heads

Tails

Heads

Tails

Tails

Heads

(-$1,$1)

($1,-$1)

($1,-$1)

(-$1,$1)

V = {v1, v2, v3, v4, v5, v6, v7}

A = {(v1, v2), (v1, v3), (v2, v4), (v2, v5), (v3, v6), (v3, v7)}r = {v1}

N = {M, S}

P0 = , PS = {v1}, and PM = {v2, v3}

(v1, v2) = (v2, v4) = (v3, v6) = Heads(v1, v3) = (v2, v5) = (v3, v7) = Tails

U1M = {v2}, U2

M = {v3}, & U1S = {v1}

g(t) = (gS(t), gM(t))g(v4) = g(v7) = (-$1, $1)g(v5) = g(v6) = ($1, -$1)

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Traditional Matching Pennies Example

v1

v3

v2

v4

v5

v6

v7

Spencer (S)

Mason (M)

Mason (M)

Heads

Tails

Heads

Tails

Tails

Heads

(-$1,$1)

($1,-$1)

($1,-$1)

(-$1,$1)

V = {v1, v2, v3, v4, v5, v6, v7}

A = {(v1, v2), (v1, v3), (v2, v4), (v2, v5), (v3, v6), (v3, v7)}r = {v1}

N = {M, S}

P0 = , PS = {v1}, and PM = {v2, v3}

(v1, v2) = (v2, v4) = (v3, v6) = Heads(v1, v3) = (v2, v5) = (v3, v7) = Tails

U1M = {v2, v3} & U1

S = {v1}

g(t) = (gS(t), gM(t))g(v4) = g(v7) = (-$1, $1)g(v5) = g(v6) = ($1, -$1)

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Contradictory Information Sets

v1

v3

v2

v4

v5

v6

v7

Spencer

Mason

Heads

Tails

Heads

Tails

Feet

Heads

(-$1,$1)

($1,-$1)

($1,-$1)

(-$1,$1)

v1

v3

v2

v4

v5

v6

v7

Spencer

Mason

Heads

Tails

Heads

Tails

Feet

Heads

(-$1,$1)

($1,-$1)

($1,-$1)

(-$1,$1)Tails

($0,$0)v8

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Example of Forgetful Information Set

v1

v3

v2

v4

v5

v6

v7

Spencer

Mason

Heads

Tails

Heads

Tails

Tails

Heads

(-$1,$1)

($1,-$1)

($1,-$1)

v8

v9

Heads

Tails

(-$1,$1)

($1,-$1)

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Another Example of Forgetfulness

Mason

Spencer Spencer

Mason

L R

L R L R

L R L R L R L R

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Strategy in Extensive Form Games

• Two Types:– Pure

– Mixed

• Pure Strategy– Compete Description of Play for All Contingencies

– Sji: set of available actions for information set Uj

i, where sji Sj

i is a particular action.

– A pure strategy is si = {s1i, s2

i,…, sk(i)i} such that si Si = S1

iS2i…Sk(i)

i.

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Examples

• Traditional Matching Pennies:– SS = S1

S = {Heads, Tails}

– SM = S1M = {Heads, Tails}

• Unfair Matching Pennies– SS = S1

S = {Heads, Tails}

– S1M = S2

M = {Heads, Tails}

– SM = {(Heads, Heads), (Heads, Tails), (Tails, Heads), (Tails, Tails)}

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Normal/Strategic/Matrix Form Games

• A strategic form game is described by: – A set N = {1,2,…,n} of players.

– A finite set of pure strategies Si for each i N where S = S1S2…Sn is the set of all possible pure strategy outcomes.

– A payoff function gi: S for each i N.

Finite Extensive form games can always be written in the strategic form!

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Example for Traditional Matching Pennies

sS

Heads Tails

sM

Heads -$1

$1 $1

-$1

Tails $1

-$1 -$1

$1

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Example for Unfair Matching Pennies

sS

Heads Tails

(Heads,Heads) -$1

$1 $1

-$1

sM

(Heads, Tails) -$1

$1 -$1

$1

(Tails,Heads) $1

-$1 $1

-$1

(Tails, Tails) $1

-$1 -$1

$1

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Classification of Games Simultaneous vs. Sequential Move

• Simultaneous Move Games:– Games where players must make choices knowing the incentives of

other players, but not their choices.

• Sequential Move Games:– Games where players make choices knowing the incentives of other

players and at least one player knows at least one choice of another before making one of his own.

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Classification of Games Perfect vs. Imperfect Information

• Perfect Information Games: – Games where all information sets contain a single node.

• Imperfect Information Games: – Games where at least one information set contains two or more nodes.

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Classification of Games Complete vs. Incomplete Information

• Complete Information Games: – Games where all players know each other’s payoffs.

• Incomplete Information Games: – Games where some players’ payoffs are unknown.