nash equilibrium - illinoishrtdmrt2/teaching/gt_2016_19/l4.pdf · nash equilibrium carlos hurtado...

Nash Equilibrium

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

Junel 10th, 2016

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda

1 Formalizing the Game

2 Pure Strategies Nash Equilibrium

3 Nash Equilibrium and Strictly Dominated Strategies

4 Existence of PSNE

5 Mixed Strategy Nash Equilibrium

6 Existence of NE

7 Exercises


Formalizing the Game

On the Agenda




4 Existence of PSNE


6 Existence of NE

7 Exercises




I Let me fix some Notation:- set of players: I = {1, 2, · · · ,N}- set of actions: ∀i ∈ I, ai ∈ Ai , where each player i has a set of actions Ai .- strategies for each player: ∀i ∈ I, si ∈ Si , where each player i has a set ofpure strategies Si available to him. A strategy is a complete contingent planfor playing the game, which specifies a feasible action of a player’sinformation sets in the game.

- profile of pure strategies: s = (s1, s2, · · · , sN) ∈∏N

i=1 Si = S.Note: let s−i = (s1, s2, · · · , si−1, si+1, · · · , sN) ∈ S−i , we will denotes = (si , s−i ) ∈ (Si , S−i ) = S.

- Payoff function: ui :∏N

i=1 Si → R, denoted by ui (si , s−i )- A mixed strategy for player i is a function σi : Si → [0, 1], which assigns aprobability σi (si ) ≥ 0 to each pure strategy si ∈ Si , satisfying∑

si∈Siσi (si ) = 1.

- Payoff function over a profile of mixed strategies:

ui (σi , σ−i ) =∑

si∈Si

∑s−i∈S−i

[∏j 6=i

σj(sj)

]σi (si )ui (si , s−i )

C. Hurtado (UIUC - Economics) Game Theory 1 / 14

Pure Strategies Nash Equilibrium

On the Agenda




4 Existence of PSNE


6 Existence of NE

7 Exercises




I Now we turn to the most well-known solution concept in game theory. We’ll firstdiscuss pure strategy Nash equilibrium (PSNE), and then later extend to mixedstrategies.

DefinitionA strategy profile s = (s1, ..., sN) ∈ S is a Pure Strategy Nash Equilibrium (PSNE) if forall i and s̃i ∈ Si , u(si , s−i ) ≥ u(s̃i , s−i ).

I In a Nash equilibrium, each player’s strategy must be a best response to thosestrategies of his opponents that are components of the equilibrium.

I Remark: Every finite game of perfect information has a Nash equilibrium (underwhat assumptions?).




I Consider the following game between 100 people. Each player selects a number, si ,between 20 and 60. Let a−i be the average of the number selected by the other 99players. That is, a−i =

∑j 6=i

sj99 .

I Let the utility of player i be ui (si , s−i ) = 100−(si − 3

2a−i)2.

I If each player maximizes his utility, the F.O.C. is:

−2(si −

32a−i

)= 0

I Hence, each player i would like to select si = 32a−i .

I Note that a−i ∈ [20, 60], hence 32a−i ∈ [30, 90].

I Then, si = 20 is dominated by si = 30. That is, regardless of the selection of theothers, si = 30 always gives more utility to player i .

I The same is true for any number between 20 and 30. Hence, each player i willselect a number between 30 and 60.

I With the same logic, a−i ∈ [30, 60], hence 32a−1 ∈ [45, 90].

I Then, playing any number below 45 is dominated by playing 45.I Appalling the same logic, all players will select 60.


Nash Equilibrium and Strictly Dominated Strategies

On the Agenda




4 Existence of PSNE


6 Existence of NE

7 Exercises




I What is the relation between strictly dominated strategies and PSNE?

TheoremEvery PSNE survives to the Iterated Delation of Strictly Dominated Strategies

Proof : Suppose otherwise. Then, in one round of Delation of Strictly DominatedStrategies we eliminate s∗, a PSNE. Suppose that for player i there exists astrategy si ∈ Si such that for all s−i ∈ S−i ,

ui (si , s−i ) > ui (s∗i , s−i ).

In particular,ui (si , s∗−i ) > ui (s∗i , s∗−i ).

Hence, s∗i is not a PSNE, which is a contradiction.




I What is the relation between strictly dominated strategies and PSNE?

TheoremIf the Iterated Delation of Strictly Dominated Strategies reaches a unique solution, thisis the unique PSNE.

Proof : Suppose that the unique solution reached by the Iterated Delation ofStrictly Dominated Strategies, s∗, is not a PSNE. That means that for some playeri there exist si such that

ui (s∗i , s∗−i ) < ui (si , s∗−i )

Then, s∗i was strictly dominated by si , which is a contradiction.

The uniqueness follows from the previous theorem.


Existence of PSNE

On the Agenda




4 Existence of PSNE


6 Existence of NE

7 Exercises


Existence of PSNE

Existence of PSNE

I Some useful tools:- Convex set: An set is convex if for every pair of points within the set, everypoint on the straight line segment that joins the pair of points is also withinthe object.Formally, let V be a vector space over the real numbers. A set S in V is saidto be convex if, for all s1 and s2 in S and all µ ∈ [0, 1], the pointσ = (1− µ)s1 + µs2 also belongs to S.

I


Existence of PSNE

Existence of PSNEI Some useful tools:

- Extreme value theorem: A real-valued function f is continuous in the closedand bounded interval [a, b], then f must attain a maximum and a minimum,each at least once.Formally, there exist numbers c and d in [a, b] such that:

f (c) ≥ f (x) ≥ f (d) for all x ∈ [a, b].

IC. Hurtado (UIUC - Economics) Game Theory 7 / 14

Existence of PSNE

Existence of PSNE

I Some useful tools:

- Compact space: For the purpose of this class we will use the Heine-Boreltheorem that states that a subset of Euclidean space is compact if and only ifit is closed and bounded.

- The extreme value theorem, also known as Weierstrass’ theorem, can begeneralized to continuous functions over compact sets:

Theorem: Suppose f is a continuous real function on a compact metric spaceX , let M = supp∈X f (p) and m = infp∈X f (p). Then, exist p and q in X suchthat f (p) = M and f (q) = m.

proof : Rudin page 89.


Existence of PSNE

Existence of PSNE

Theorem

(Existence of PSNE). Suppose each Si ⊂ RN is non-empty, compact and convex; andeach ui : S → R is continuous in s and quasi-concave in si . Then there exists a PSNE.

I A finite strategy profile space, S, cannot be convex (why?), so this existenceTheorem is only useful for infinite games.

I To prove this we use Kakutani’s Fix Point Theorem over correspondences. This isout of the scope of this class. You will learn it an advanced course (go for theMaster or the Ph.D.!!).

I quasi-concavity a player’s utility function plays a key role.


Mixed Strategy Nash Equilibrium

On the Agenda




4 Existence of PSNE


6 Existence of NE

7 Exercises




I It is straightforward to extend our definition of Nash equilibrium to this case, andthis includes the earlier definition of PSNE.

DefinitionA strategy profile σ = (σ1, · · · , σN) ∈ ∆(S) is a Nash equilibrium if for all i andσ̃i ∈ ∆(Si ), ui (σi , σ−i ) ≥ ui (σ̃i , σ−i ).

I We call this Mixed Strategy Nash Equilibrium (MSNE).

I To see why considering mixed strategies is important, observe that MatchingPennies (version A) has no PSNE, but does have MSNE.




I Matching Pennies (version A).

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

Rules: Each player simultaneously puts a penny down, either heads up or tails up.

Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,player 2 pays 1 dollar to player 1.

I Each player randomizes over H and T with equal probability.

I In fact, when player i behaves in this way, player j 6= i is exactly indifferentbetween playing H or T .

I That is, in the MSNE, each player who is playing a mixed strategy is indifferentamongst the set of pure strategies he is mixing over.

I This remarkable property is very general and is essential in helping us solve forMSNE in many situations.


Existence of NE

On the Agenda




4 Existence of PSNE


6 Existence of NE

7 Exercises


Existence of NE

Existence of NE

Theorem(Existence of NE). Every finite game has a Nash equilibrium (possibly in mixedstrategies).

I Proof. For each i , given the finite space of pure strategies, Si , the space of mixedstrategies, ∆(Si ), is a (non-empty) compact and convex subset of R|Si |. The utilityfunctions ui : ∆(S)→ R defined by

ui (σi , σ−i ) =∑si∈Si

∑s−i∈S−i

[∏j 6=i

σj(sj)

]σi (si )ui (si , s−i )

are continuous in σi and quasi-concave in σi . Thus, the previous Theorem impliesthat there is a pure strategy Nash equilibrium of the infinite normal-form gameI,∆(Si ), ui ; this profile is a (possibly degenerate) mixed strategy Nash equilibriumof the original finite game.

I The critical need to allow for mixed strategies is that in finite games, the purestrategy space is not convex, but allowing players to mix over their pure strategies"convexifies" the space.


Exercises

On the Agenda




4 Existence of PSNE


6 Existence of NE

7 Exercises


Exercises

Exercises

I Rock (R), Paper (P) or Scissors (S) is a zero sum game. A player who decides toplay rock will beat another player who has chosen scissors ("rock crushes scissors")but will lose to one who has played paper ("paper covers rock") (?); a play ofpaper will lose to a play of scissors ("scissors cut paper"). The reduced form ofthis game is:

I

1/2 R P SR 0,0 -1,1 1,-1P 1,-1 0,0 -1,1S -1,1 1,-1 0,0

Find the Nash Equilibrium of this game.


Exercises

Exercises

I Find the PSNE and the MSNE in the following games:

I


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