Week 14 Game Theory (Jehle and Reny, Ch.7 Mas-Colell et al ......Week 14 Game Theory (Jehle and Reny, Ch.7 Mas-Colell et al.,Ch.7-8 Watson) Serçin ahin ld zY echnicalT University

Introduction Basic Elements of Noncooperative Games Analyzing Behaviour in Static Settings

Week 14Game Theory

(Jehle and Reny, Ch.7Mas-Colell et al.,Ch.7-8

Watson)

Serçin �ahin

Y�ld�z Technical University

25 December 2012


Interdependence means that one person's behaviour a�ects

another person's well-being, either positively or negatively.

Situations of interdependence are called strategic settings

because, in order for a person to decide how best to behave,

he must consider how others around him choose their actions.

Games are formal descriptions of strategic settings. Thus,

game theory is a methodology of formally studying situations

of interdependence.


The Penalty Kick Game


Game theory analysis consists of two major subbranches:

The noncooperative framework treats all of the agents' actionsas individual actions. An individual action is something that aperson decides on his own, independently of the other peoplepresent in the strategic environment.Analysing behaviour in models with joint actions requires adi�erent set of concepts from those used for noncooperativeenvironments; this alternative theory is called cooperative

game theory.


There are three major tensions of strategic interaction that are

identi�ed by the theory:

1 the con�ict between individual and group interests2 strategic uncertainty3 ine�cient coordination


To describe a situation of strategic interaction with a game,

we need to know the following �ve elements that these

representations have in common:

1 A list of players: Who is involved?2 The rules of the game: A complete description of what the

players can do (their possible actions)3 A description of what the players know when they act4 A speci�cation of how the players' actions lead to outcomes:

For each possible set of actions by the players, what is theoutcome of the game?

5 A speci�cation of the players' preferences over outcomes


Example 1: Chess

1 Players: There are two players

2 Rules: The players alternate in moving pieces on the game

board, subject to rules about what moves can be made in any

given con�guration of the board

3 Information: players observe each other's moves, so each

knows the entire history of play as the game progresses

4 Outcomes: a player who captures the other player's king wins

the game; in certain situations a draw is decleared.

5 Preferences over outcomes: Players prefer winning over a draw

and a draw over losing.


Example 2: Matching Pennies

1 Players: There are two players, denoted 1 and 2

2 Rules: Each player simultaneously puts a penny down, either

heads up or tails up.

3 Information: Both player observes his own and opponent's

penny at the same time.

4 Outcomes: If the two pennies match (either both heads up or

both tails up) player 1 pays 1 dollar to player 2; otherwise,

player 2 pays 1 dollar to player 1.

5 Preferences over outcomes: Player 1 has a preference for the

pennies do not match, and Player 2 has a preference for the

pennies do match.


The Extensive Form

The extensive form relies on the conceptual apparatus known

as a game three.

Nodes represent places where something happens in the game

(such as a decision by one of the players)

Branches indicate the various actions that players can choose.

We represent nodes by solid circles and branches by arrows

connecting the nodes.


The game starts with a node, and this node is called the initial

node.

Other nodes in the game are called decision nodes, because

players make decisions at these places in the game.

The other nodes are called end nodes; they represent

outcomes of the game - places where the game ends.

Each end node also corresponds to a unique path through the

tree, which is a way of getting from the initial node through

the tree by following branches in the direction of the arrows.

Each end node is also followed by a vector of payo�s.


Matching Pennies Version B


It is common to use the term information set to specify the

players' information at decision nodes in the game.

Formally, the elements of an information set are a subset of a

particular player's decision nodes. The interpretation is that

when play has reached one of the decision nodes in the

information set and it is player's turn to move, she does not

know which of these nodes she is actually at.

If one player cannot distinguish between some nodes, then, his

lack of information is shown with a dashed line connecting

these nodes.

Every decision node is contained in an information set; some

information sets singleton -consist of only one node.

Extensive form games in which every information set is a

singleton are called games of perfect information.All other

games are called games with imperfect information.



Perfect RecallPerfect recall means that a player does not forget what she once

knew, including her own actions.


In an extensive form, we must draw one player's decision before

that of the other, but it is important to realize that this does not

necessarily correspond to the actual timing of the strategic setting.


Up to this point, the outcome of a game has been a

deterministic function of the players' choices.

In many games, however, there is an element of chance.

This, too, can be captured in the extensive form representation

by including random moves of nature.

Its moves are assumed to be made according to a �xed

probability distribution.


Gift Game


Mathematical Representation of Extensive Form:Formally, a game represented in extensive form, denoted by Γ ,

consists of the following items:

A �nite set of players, N.

A set of actions, A, which includes all possible actions that

might potentially be taken at some point in the game. A need

not be �nite.


A set of nodes or histories, X, where

X contains a distinguished element, x0, called the initial node,or empty historyeach x ∈ X\{x0} takes the form x = (a1, a2, ..., ak) for some�nitely many actions ai ∈ A, andif (a1, a2, ..., ak) ∈ X\{x0} for some k > 1, then(a1, a2, ..., ak−1) ∈ X\{x0}.A node, or history, is then simply a complete description of theactions that have been taken so far in the game. We shall usethe terms history and node interchangeably. For futurereference let

A(x) ≡ {a ∈ A | (x , a) ∈ X}denote the set of actions available to the player whose turn itis to move after the history x ∈ X\{x0}.


A set of actions, A(x0) ⊆ A, and a probability distribution, π,on A(x0) to describe the role of chance in the game. Chance

always moves �rst, and just once, by randomly selecting an

action from A(x0) using the probability distribution π. Thus,(a1, a2, ..., ak) ∈ X\{x0} implies that ai ∈ A(x0) for i = 1 and

only i = 1.

A set of end nodes, E ≡ {x ∈ X | (x , a) /∈ X for all a ∈ A}.Each end node describes one particular complete play of the

game form beginning to end.

A function, ι : X\(E ∪ {x0}) −→ N that indicates whose turn

it is at each decision node in X. For future reference, let

Xi ≡ {x ∈ X\(E ∪ {x0}) | ι(x) = i}denote the set of decision nodes belonging to player i .


A partition, I , of the set of decision nodes, X\(E ∪ {x0}), suchthat if x and x

′are in the same element of the patition then

(i) ι(x) = ι(x′), and

(ii)A(x) = A(x′).

For future reference, let

Ii ≡ {I (x) | ι(x) = i , some x ∈ X\(E ∪ {x0})}denote the set of information sets belonging to player i .


For each i ∈ N, a von Neumann-Morgenstern payo� function

whose domain is the set of end nodes, ui :;E −→ <. Thisdescribes the payo� to each player for every possible complete

play of the game.

We write Γ =< N,A,X ,E , ι, π, I , (ui )i∈N >. If the sets of actions,

A, and nodes, X , are �nite, then Γ is called a �nite extensive form

game.



The buyer and seller of the used car.1 Again there are two players, so N = {S ,B}, where S denotes

seller, and B , buyer.2 To keep things simple, assume that the seller, when choosing a

price, has only two choices: high and low. The set of actions

that might arise is A = { repair, don't repair, price high, price

low, accept, reject }.3 Because chance plays no role here, rather than give it a single

action, we simply eliminate chance from the analysis.4 A node in this game is, for example, x = { repair, price high }.

At this node, x , it is the buyer's turn to move, so that

ι(x) = B .5 Because at this node, the buyer is informed of the price chosen

by the seller, but not of the seller's repair decision, I (x) = {(repair, price high ), ( don't repair, price high )}. That is, whennode x is reached, the buyer is informed only that one of the

two histories in I (x) has occured; he is not informed of which

one, however.


Strategy

Formally, a (pure) strategy for player i in an extensive form

game Γ is a function si : Ii −→ A, satisfying si (I (x)) ∈ A(x)for all x with ι(x) = i .

Let Si denote the set of (pure) strategies (also called strategy

space)for player i in Γ. That is, Si is a set of comprising each of

the possible strategies of player i in Γ. We shall assume that Γis a �nite game. Consequently, each of the sets Si is also �nite.


Strategies in the Matching Pennies Version B


Strategies in the Matching Pennies Version B

Strategy 1(s1): Play H if player 1 plays H; play H if player 1

plays T.

Strategy 2(s2): Play H if player 1 plays H; play T if player 1

plays T.

Strategy 3(s3): Play T if player 1 plays H; play H if player 1

plays T.

Strategy 4(s4): Play T if player 1 plays H; play T if player 1

plays T.


A strategy pro�le (or joint strategy) is a vector of strategies,

one for each player. In other words, a strategy pro�le describes

strategies for all of the players in the game. For example,

suppose we are studying a game with n players. Then a typical

strategy pro�le is a vector s = (s1, s2, s3, ..., sn) where si is thestrategy of player i , for i = 1, 2, 3, ..., n.

We will also sometimes separate the strategy pro�le s into the

strategy of player i and the strategies of the other players, we

write s = (si , s−i ). s−i is the (N − 1) vector of strategies for

players other than i : s−i = (s1, s2, ..., si−1, si+1, ..., sn).

Let S denote the set of strategy pro�les. Mathematically,

S = S1 × S2 × ...× Sn.


The Strategic (Normal) Form GameA strategic (normal) form game is a tuple G = (Si , ui )

Ni=1 where for

each player i = 1, 2, ...N, Si is the set of strategies available to

player i , and ui : ×Nj=1Sj −→ R describes player i 's payo� as a

function of the strategies chosen by all players.

A strategic form game is �nite if each player's strategy set contains

�nitely many elements.


Matching Pennies Version B in the Normal Form


It is clear that for any extensive form representation of a game,

there is a unique normal form representation. The converse is not

true, however. Many di�erent extensive forms may be represented

by the same normal form.


Beliefs

Mathematically, a belief of player i is a probability distribution

over the strategies of the other players. We denote such a

probability distribution ∆µ−i and write ∆µ−i ∈ ∆S−i , where∆S−i is the set of probability distributions over the strategies

of all the players except player i .

The belief of player i about the behaviour of player j is afunction µj ∈ ∆Sj such that, for each strategy sj ∈ Sj ofplayer j , µj(sj) is interpreted as the probability that player ithinks player j will play sj .As a probability distribution, µj has the property that

µj(sj) ≥ 0 for each sj ∈ Sj , and∑

sj∈Sj µj(sj) = 1.


Mixed Strategies

A mixed strategy for a player is the act of selecting a strategy

according to a probability distribution.

For a �nite strategic form game G = (Si , ui )Ni=1, a mixed

strategy mi for player i is a probability distribution over Si .That is, mi : Si −→ [0, 1] assigns to each si ∈ Si theprobability, mi (si ), that si will be played.

We shall denote the set of mixed strategies for player i by Mi .

Consequently, Mi = {mi : Si −→ [0, 1] |∑

si∈Si mi (si ) = 1}.Suppose that player i has M pure strategies in set

Si = {s1i , s2i , ..., sMi}.


Then, this set forms a unit simplex on the M-dimensional

Euclidian space and it is called the mixed extension of Si .∆(Si ) = {(m1i , ...,mMi ) ∈ RM : mji > 0 for all j = 1, ...,M

and∑M

j=1mji = 1}.Let M = ×N

i=1Mi denote the set of joint mixed strategies.

From now on, we shall drop the word "mixed" and simply call

m ∈ M a joint strategy and mi ∈ Mi a strategy for player i .


When players randomize over their pure strategies, the induced

outcome is itself random, leading to a probability distribution

over the terminal nodes of the game.

Let us assume that the players' payo�s are in fact von

Nuemann-Morgenstern utilities, and that they will behave to

maximise their expected utility.

If ui is a von Neumann-Morgenstern utility function on S, and

the strategy m ∈ M is played then player i 's expected utility is

ui (m) ≡∑

s∈S m1(s1)...mN(sN)ui (s)


Prisoners' Dilemma


Strictly Dominant Strategies

Let S = S1 × ...× SN denote the set of strategy pro�les. The

symbol, −i , denotes all players except player i. So, forexample, s−i denotes an element of S−i , which itself denotes

the set S1 × ...× Si−1 × Si+1 × ...× SN . Then we have the

following de�nition:

A strategy, si , for player i is strictly dominant if

ui (si , s−i ) > ui (si , s−i ) for all (si , s−i ) ∈ S with si 6= si .


Strictly Dominated Strategies

We should expect that player i will not play dominated

strategies, those for which there is some alternative strategy

that yields him a greater payo� regardless of what the other

players do.

Player i 's strategy si strictly dominates another of his

strategies si , for all s−i ∈ S−i . In this case, we also say that siis strictly dominated in S .




Iteratively Eleminating Strictly Dominated Strategies

Let S0i = Si for each player i , and for n ≥ 1, let Sn

i denote

those strategies of player i surviving after the nth round of

elimination. That is si ∈ Sni if si ∈ Sn−1

i is not strictly

dominated in Sn−1.

A strategy si for player i is iteratively strictly undominated in

S (or survives iterative elimination of strictly dominated

strategies) if si ∈ Sni , ∀n ≥ 1.

Once we have determined the set of undominated pure

strategies for player i , we need to consider which mixed

strategies are undominated.


Weakly Dominated Strategies

So far, we have considered only notions of strict dominance.

Related notions of weak dominance are also available.

Player i 's strategy si weakly dominates another of his

strategies si , if

ui (si , s−i ) ≥ ui (si , s−i ) for all s−i ∈ S−i ,

with at least one strict inequality. In this case, we also say that

si is weakly dominated in S .


Iteratively Eliminating Weakly Dominated Strategies

Let W 0i = Si for each player i , and for n ≥ 1, let W n

i denote

those strategies of player i surviving after the nth round of

elimination of weakly dominated strategies.

That is, si ∈W ni if si ∈W n−1

i is not weakly dominated in

W n−1 = W n−11 × ...×W n−1

N .

A strategy si for player i is iteratively weakly undominated in S(or survives iterative elimination of weakly dominated

strategies) if si ∈W ni for all n ≥ 1.



Best Response and Rationalizability

The set of rationalizable strategies consists precisely of those

strategies that may be played in a game where the structure of

the game and the players' rationality are common knowledge

among the players.

To maximize the payo� that you expect to obtain -which we

assume is the mark of rational behaviour- you should select the

strategy that yields the greatest expected payo� against your

belief. Such a strategy is called a best response.

Formally,suppose player i has a belief µ−i ∈ ∆S−i about thestrategies played by the other players. Player i 's strategysi ∈ Si is a best response if

ui (si , µ−i ) ≥ ui (s′i , µ−i )

for every s′i ∈ Si .


In a �nite game, every belief has at least one best response.

For each belief µ−i of player i , we denote the set of best

responses by BRi (µ−i ).In this way, the belief rationalizes

playing the strategy. Furthermore, each player should assign

positive probability only to strategies of the other players that

can be similarly rationalized.

Clearly, a player should not play a strategy that is never a best

response.


As in the case of strictly dominated strategies, common

knowledge of rationality and the game's structure implies that

we can iterate the deletion of strategies that are never best

response.

Equally important, the strategies that remain after this

iterative deletion are the strategies that a rational player can

justify, or rationalize.The set of strategies surviving this

iterative deletion process can be said to be precisely the set of

strategies that can be played by rational players in a game in

which the players' rationality and the structure of the game are

common knowledge. They are known as rationalizable

strategies.


Dominance and Best Response Compared

We have said that the set of rationalizable strategies is no

larger than the set remaining after iterative deletion of strictly

dominated strategies.

It turns out, however, that for the case of two-player games

(N = 2), these two sets are identical because in two-player

games a (mixed) strategy mi is a best response to some

strategy choice of a player's rival whenever mi is not strictly

dominated. Then since this process is equivalent to iterated

dominance, we can just perform the iterative deletion of

dominated strategies.

With more than two players, however, there can be strategies

that are never a best response and yet are not strictly

dominated.


Nash Equilibrium

In the strategic setting, just as in the demand-supply setting,

regularities in behaviour that can be rationally sustained will

be called equilibria.

If the agreement is to play strategy pro�le s and si /∈ BRi (s−i )for some player i , then this player has no incentive to abide by

the agreement and will chose a strategy that is di�erent from

si . Thus, a Nash equilibrium describes behaviour that can be

rationally sustained.

Formally, given a strategic form game G = (Si , ui )Ni=1, the

strategy pro�le s ∈ S is a pure strategy Nash equilibrium of Gif for each player

i , ui (s) ≥ ui (si , s−i )

for all si ∈ Si .


A compact restatement of the de�nition of a Nashequilibrium

Lets de�ne for each player i , a pure-strategy best-response

correspondence ψi : S → Si which speci�es for every strategy

pro�le m−i ∈ M−i by player i 's opponents a set

BRi (m−i ) ⊂ Si of player i 's pure strategies which are best

responses.

Now we form a new correspondence ψ by forming the

Cartesian product of the n personal best response

correspondences ψi . We de�ne for every strategy pro�le s ∈ S ,

ψ(s) = ×i∈Nψi (s)

Therefore, we see that ψ is a correspondence itself from the

space of strategy pro�les into the space of strategy pro�les;

i.e., ψ : S → S .


In a Nash equilibrium, each player i 's strategy si is a best

response to the other players' strategies si . Because this

inclusion must hold for all players, we have

(s1, ..., sn) ∈ ×i∈Nψi (s) = ψ(s)

Then, a Nash equilibrium pro�le is a �xed point of the best

response correspondence ψ. This logic is reversible: any �xed

point of the best response correspondence is a Nash

equilibrium pro�le.





Mixed Strategy Nash Equilibrium

It is straightforward to extend the de�nition of Nash

equilibrium to games in which we allow the players to

randomize over their pure strategies.

Given a �nite strategic form game G = (Si , ui )Ni=1, a strategy

pro�le m ∈ M is a Nash equilibrium of G if for each player i ,ui (m) ∈ M is a Nash equilibrium of G if for each player i ,

ui (m) ≥ ui (mi , m−i ) for all mi ∈ Mi .

A necessary and su�cient condition for mixed strategy pro�le

m to be a Nash equilibrium of game G is that each player,

given the distribution of strategies played by his opponents, is

indi�erent among all the pure strategies that he plays with

positive probability.



Existence of Nash EquilibriumEvery �nite strategic form game possesses at least one Nash

equilibrium.


References

Jehle and Reny, 2011, Advanced Microeconomic Theory, Third

Edition, Prentice Hall

Mas Colell, Whinston and Green, 1995, Microeconomic

Theory, Oxford University Press

Watson, 2002, Strategy: An Introduction to Game Theory,

W.W.Norton and Company

Documents

Week 14 Game Theory (Jehle and Reny, Ch.7 Mas-Colell et al ......Week 14 Game Theory (Jehle and Reny, Ch.7 Mas-Colell et al.,Ch.7-8 Watson) Serçin ahin ld zY echnicalT University