Lecture 5: Extensive Form Games with Perfect...

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Lecture 5: Extensive Form

Games with Perfect Information

INSE 6441/2 DD - Applied Game

Theory and Mechanism Design

Fall 2014Acknowledgement: Jackson, Leyton-Brown & Shoham

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Overview

Informal Treatment

Formalization

Best Response and Nash Equilibrium

Subgame Perfection

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Ω Sequential Move Games

Sequential-move games entail strategic

situations in which time is considered and

there is a strict order of play

Players take turns making their moves, and

they know what other players have done

To play well in such games, participants must

use a particular type of interactive thinking

Each player must consider

If I make this move, how will my opponent

respond?

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Ω Sequential Move Games

To make decisions, players need to consider

how their current actions will influence future

actions, both for their rivals and for

themselves

Players thus decide their current moves on

the basis of calculations of future

consequence

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Ω Game Trees

Game trees provide a graphical technique for

displaying and analyzing sequential-move-games

Game trees are referred to as extensive form games

They are very similar to decision trees

Decision trees show all the successive decision

points, or nodes, for a singled decision maker in a

neutral environment

Decision trees also include branches corresponding

to the available choices emerging from each node

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Ω Game Trees

Game trees are joint decision trees for

all of the players in a game

The trees illustrate all of the possible

actions that can be taken by all of the

players and indicate all of the possible

outcomes from the game

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Initial node or

root

Decision

Nodes

Terminal nodes

Branches

Payoffs

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Example

Ω Example: Uncertainty

If Ann chooses "Go" and then Chris chooses "Risky,"

something happens at random

a fair coin is tossed and the outcome of the game is

determined by whether that coin comes up "heads" or "tails"

This aspect of the game is handled by introducing an

outsider player called "Nature"

It chooses one of two branches each with 50%

probability

The probabilities here are fixed by the type of random

event, a coin toss, but could vary in other

circumstances

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ΩExample: Payoffs

We show the outcome of each sequence of

actions, as the payoffs of the players

For our four players, we list the payoffs in the

alphabetical order (Ann, Bob, Chris, Deb)

The inclusion of a random event (a choice made

by Nature) means that players need to determine

what they get on average:

Expected Payoffs. Example: Ann’s

expected payoff = (0.5X6+0.5X2) = 4

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Ω Strategies

Move: a single action taken by a player at

a node

Players should make plans for the

succession o f moves that they expect to

make in all of the various eventualities that

might arise in the course of a game

Such a plan of action is called a strategy

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Ω Strategies

In the tree example, Bob, Chris, and Deb

each get to move at most once

For them, there is no distinction between a move

and a strategy

But Ann has two opportunities to move, so

her strategy needs a fuller specification

One strategy for her is, "Choose Stop, and

then if Bob chooses 1, choose Down."

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Solving The Games

Take, for example, a teenager named Carmen who

is deciding whether to smoke

First, she has to decide whether to try smoking at all

If she does try it, she has the further decision of

whether to continue

We illustrate this example as a simple decision tree

The solution is, she should try but not continue

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We convert the decision tree of the previous example

into a game tree, by distinguishing between the two

players who make the choices at the two nodes

At the initial node, "Today's Carmen" decides whether

to try smoking

If her decision is to try, then the addicted "Future

Carmen" comes into being and chooses whether to

continue or not

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Solving The Games

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Solving The Games


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Solution in this case is:

Today’s Carman should not try smoking

Future Carman should continue smoking

Solving The Games

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Another Example

If Instead of Emily, Nina or Talia moved first, what

would happen?

The advantage is given to the first mover

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First and Second Mover

Emily gets her best outcome, because she can take

advantage of the opportunity to make the first move

Such first-mover advantage does not exist in all games

Second-mover advantage is possible

Example: Product Price (Catalog ) between competing

companies

First-mover advantage comes from the ability to commit

oneself to an advantageous position and to force the

other players to adapt to it

second-mover advantage comes from the flexibility to

adapt oneself to the others‘ Choice

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Ω

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Overview

Informal Treatment

Formalization


Subgame Perfection


Ω Formal Definition

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Formal Definition


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Formal Definition

Ω Example: The Sharing Game

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Ω Pure Strategy

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A pure strategy for a player in a perfect-

information game is a complete specification of

which action to take at each node belonging to

that player.

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Player 1 has 3 pure strategies

Player 2 has 8 pure strategies

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Pure Strategy


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Pure strategies for player 2:

S2 = CE, CF, DE, DF

Pure strategies for player 1:

S1 = BG, BH, AG, AH

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Example 2

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Overview

Informal Treatment

Formalization


Subgame Perfection


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Given our new definition of pure

strategy, we are able to reuse our old

definitions of:

Mixed strategies

Best response

Nash equilibrium

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Nash Equilibrium

ΩInduced Normal Form

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Any extensive form game can be

converted into a normal form game


Ω Induced Normal Form

This is easy to see, since the players move

sequentially

It is important to notice that we cannot always

perform the transformation from normal form

to extensive form

Matching pennies cannot be written as a perfect-

information extensive form game

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ΩInduced Normal Form

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Pure strategy equilibria:

AG, CF

AH, CF

BH, CE


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Overview

Informal Treatment

Formalization


Subgame Perfection

ΩDiscussing Equilibrium

There’s something intuitively wrong with the equilibrium

BH, CE

Why would player 1 ever choose to play H if he got to

the second choice node?

After all, G dominates H for him

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Discussing Equilibrium

Player 1 does select H to threaten player 2, to

prevent him from choosing F, and so gets 5

However, this seems like a non-credible threat

If player 1 reached his second decision node, would

he really follow through and play H?

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Ω

Formal Definitions

s is a subgame perfect equilibrium (SPE) of G iff for any

subgame G′ of G, the restriction of s to G′ is a Nash

equilibrium of G′

Since G is its own subgame, every SPE is a NE.

This definition rules out “non-credible threats”

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Example

AG, CF: is subgame perfect

BH, CE: BH is non-credible; not subgame perfect

AH, CF: AH is non-credible, though H is “off-path”

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