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### Introduction

APEC 8205: Applied Game Theory

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

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.

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!

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.

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.

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!

Describing Extensive Form Games

- Elements of a Game Tree:
- V: Set of Nodes
- A: Set of branches
- r: Root Node

Example Game Tree

v4

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

v2

A = {(v1, v2), (v1, v3), (v2, v4),

(v2, v5), (v3, v6), (v3, v7)}

v5

v1

v6

r = {v1}

v3

v7

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.

Examples

Example Path = v1, v2,v4

v4

v2

Not a Path = v1, v2,v6

v5

For v2, (v1, v2) is incoming, while

(v2, v4) and (v2, v5) are outgoing.

v1

v6

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

v3

v7

Decision Nodes = {v1, v2,v3}

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, U2i, …, Uk(i)i.
- describe an n-dimensional vector, g(t) = (g1(t), g2(t),…, gn(t)), of payoffs for each terminal node t.

Unfair Matching Pennies Example

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

v4

(-$1,$1)

A = {(v1, v2), (v1, v3), (v2, v4),

(v2, v5), (v3, v6), (v3, v7)}

Heads

Mason (M)

v2

r = {v1}

Heads

N = {M, S}

v5

Tails

Spencer (S)

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

($1,-$1)

v1

(v1, v2) = (v2, v4) = (v3, v6) = Heads

(v1, v3) = (v2, v5) = (v3, v7) = Tails

v6

Heads

($1,-$1)

Tails

U1M = {v2}, U2M = {v3}, & U1S = {v1}

v3

Mason (M)

g(t) = (gS(t), gM(t))

g(v4) = g(v7) = (-$1, $1)

g(v5) = g(v6) = ($1, -$1)

v7

Tails

(-$1,$1)

Traditional Matching Pennies Example

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

v4

(-$1,$1)

A = {(v1, v2), (v1, v3), (v2, v4),

(v2, v5), (v3, v6), (v3, v7)}

Heads

Mason (M)

v2

r = {v1}

Heads

N = {M, S}

v5

Tails

Spencer (S)

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

($1,-$1)

v1

(v1, v2) = (v2, v4) = (v3, v6) = Heads

(v1, v3) = (v2, v5) = (v3, v7) = Tails

v6

Heads

($1,-$1)

Tails

U1M = {v2,v3} & U1S = {v1}

v3

Mason (M)

g(t) = (gS(t), gM(t))

g(v4) = g(v7) = (-$1, $1)

g(v5) = g(v6) = ($1, -$1)

v7

Tails

(-$1,$1)

Contradictory Information Sets

v4

v4

(-$1,$1)

(-$1,$1)

Heads

Heads

Mason

Mason

v2

v2

Heads

Heads

v5

v5

Tails

Tails

Spencer

Spencer

($1,-$1)

($1,-$1)

v1

v1

v6

v6

Heads

($1,-$1)

Heads

($1,-$1)

Tails

Tails

v8

Feet

($0,$0)

v3

v3

v7

v7

Feet

Tails

(-$1,$1)

(-$1,$1)

Example of Forgetful Information Set

v4

(-$1,$1)

Heads

Mason

v2

Heads

v5

Tails

Spencer

($1,-$1)

v1

v6

Heads

($1,-$1)

Tails

v8

v3

(-$1,$1)

Heads

Tails

v7

v9

Tails

($1,-$1)

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 Uji, where sjiSji is a particular action.
- A pure strategy is si = {s1i, s2i,…, sk(i)i} such that siSi = S1iS2i…Sk(i)i.

Examples

- Traditional Matching Pennies:
- SS = S1S = {Heads, Tails}
- SM = S1M = {Heads, Tails}
- Unfair Matching Pennies
- SS = S1S = {Heads, Tails}
- S1M = S2M = {Heads, Tails}
- SM = {(Heads, Heads), (Heads, Tails), (Tails, Heads), (Tails, Tails)}

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 iN where S = S1S2…Sn is the set of all possible pure strategy outcomes.
- A payoff function gi: S for each iN.

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

Classification of GamesSimultaneous 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.

Classification of GamesPerfect 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.

Classification of GamesComplete 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.

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