Static Games: Rationalizable Strategies and

Carlos Hurtado

Department of Economics University of Illinois at Urbana-Champaign [email protected]

Jun 21th, 2017

C. Hurtado (UIUC - Economics) On the Agenda

1 Formalizing the Game

2

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory Formalizing the Game On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Strategy Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory Formalizing the Game Formalizing the Game

I Let us 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 of pure strategies Si available to him. A strategy is a complete contingent plan for playing the game, which specifies a feasible action of a player’s information sets in the game. QN - profile of pure strategies: s = (s1, s2, ··· , sN ) ∈ i=1 Si = S.

Note: let s−i = (s1, s2, ··· , si−1, si+1, ··· , sN ) ∈ S−i , we will denote s = (si , s−i ) ∈ (Si , S−i ) = S. QN - Payoff function: ui : 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 a probability σi (si ) ≥ 0 to each pure strategy si ∈ Si , satisfying P σi (si ) = 1. si ∈Si

C. Hurtado (UIUC - Economics) Game Theory 1 / 25 Formalizing the Game Formalizing the Game

I Notice now that when players use (nondegenerate) mixed strategies, this induces a probability distribution over terminal nodes of the game.

I But we can easily extend payoffs again to define payoffs over a profile of mixed strategies as follows:

X ui (σ1, ··· , σN ) = [σ1(s1) ··· σN (sN )] ui (s1, ··· , sN ) s∈S " # X X Y ui (σi , σ−i ) = σj (sj ) σi (si )ui (si , s−i )

si ∈Si s−i ∈S−i j6=i

I For the above formula to make sense, it is critical that each player is randomizing independently. That is, each player is independently tossing her own die to decide on which pure strategy to play.

C. Hurtado (UIUC - Economics) Game Theory 2 / 25 Formalizing the Game Formalizing the Game

I If si is a strictly dominant strategy for player i, then for all σi ∈ ∆(Si ), σi 6= si , and all σ−i ∈ ∆(S−i ), ui (si , σ−i ) > ui (σi , σ−i ).

I Let σi ∈ ∆(Si ), with σi 6= si , and let σ−i ∈ ∆(S−i ). Then, " # X Y ui (si , σ−i ) = σj (sj ) ui (si , s−i )

s−i ∈S−i j6=i and " # X X Y ui (σi , σ−i ) = σj (sj ) σi (˜si )ui (˜si , s−i )

˜si ∈Si s−i ∈S−i j6=i

Then, ui (si , σ−i ) − ui (σi , σ−i ) is !" # X Y X σj (sj ) ui (si , s−i ) − σi (˜si )ui (˜si , s−i )

s−i ∈S−i j6=i ˜si ∈Si

C. Hurtado (UIUC - Economics) Game Theory 3 / 25 Formalizing the Game Formalizing the Game

I ui (si , σ−i ) − ui (σi , σ−i ) is !" # X Y X σj (sj ) ui (si , s−i ) − σi (˜si )ui (˜si , s−i )

s−i ∈S−i j6=i ˜si ∈Si

I Since si is strictly dominant, ui (si , s−i ) > ui (˜si , s−i ) for all ˜si 6= si and all s−i .

X I Hence, ui (si , s−i ) > σi (˜si )ui (˜si , s−i ) for any σi ∈ ∆(Si ) such that σi 6= si ˜si ∈Si (why?).

I This implies the desired inequality: ui (si , σ−i ) − ui (σi , σ−i ) > 0

C. Hurtado (UIUC - Economics) Game Theory 4 / 25 Formalizing the Game Formalizing the Game

I We learned that: If si is a strictly dominant strategy for player i, then for all σi ∈ ∆(Si ), σi 6= si , and all σ−i ∈ ∆(S−i ), ui (si , σ−i ) > ui (σi , σ−i ).

I Exercise 1. Show that there can be no strategy σi ∈ ∆(Si ) such that for all si ∈ Si and s−i ∈ S−i , ui (σi , s−i ) > ui (si , s−i ).

I The preceding Theorem and Exercise show that there is absolutely no loss in restricting attention to pure strategies for all players when looking for strictly dominant strategies.

C. Hurtado (UIUC - Economics) Game Theory 5 / 25 Rationalizability On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Strategy Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory Rationalizability Rationalizability

Penalty Kick Game

l r L 4,-4 9,-9 M 6,-6 6,-6 R 9,-9 4,-4

I Penalty Kick Game is one of the most important games in the world.

I This game has no dominant strategies.

I We need refinements to solve more games.

C. Hurtado (UIUC - Economics) Game Theory 6 / 25 Rationalizability Rationalizability

l r

9 E[L]

8

7

6 E[M]

5

4 E[R]

3

2

1

I Do not shoot to the middle I Do not use a strategy that is never a

C. Hurtado (UIUC - Economics) Game Theory 7 / 25 Rationalizability Rationalizability

Definition

A strategy σi ∈ ∆(Si ) is a best response to the strategy profile σ−i ∈ ∆(S−i ) if u(σi , σ−i ) ≥ u(˜σi , σ−i ) for all σ˜i ∈ ∆(Si ). A strategy σi ∈ ∆(Si ) is never a best response if there is no σ−i ∈ ∆(S−i ) for which σi is a best response.

I The idea is that a strategy, σi , is a best response if there is some strategy profile of the opponents for which σi does at least as well as any other strategy.

I Conversely, σi is never a best response if for every strategy profile of the opponents, there is some strategy that does strictly better than σi .

I Clearly, in any game, a strategy that is strictly dominated is never a best response.

I Exercise 2. Prove that in 2-player games, a pure strategy is never a best response if and only if it is strictly dominated.

C. Hurtado (UIUC - Economics) Game Theory 8 / 25 Rationalizability Rationalizability

I In games with more than 2 players, there may be strategies that are not strictly dominated that are nonetheless never best responses. I As before, it is a consequence of ”rationality” that a player should not play a strategy that is never a best response. That is, we can delete strategies that are never best responses. I By iterating on the knowledge of rationality, we iteratively delete strategies that are never best responses. I The set of strategies for a player that survives this iterated deletion of never best responses is called her set of rationalizable strategies. I The rationalizable actions can be computed as follows: 1 Start with the full action set for each player. 2 Remove actions which are never a best responses to any belief about the opponents’ actions. 3 Repeat process with the opponents’ remaining actions until no further actions are eliminated. 4 In this process leaves a non-empty set of actions for each player those are the rationalizable actions.

C. Hurtado (UIUC - Economics) Game Theory 9 / 25 Rationalizability Rationalizability

Definition

I σi ∈ ∆(Si ) is a 1-rationalizable strategy for player i if it is a best response to some strategy profile σ−i ∈ ∆(S−i ). I σi ∈ ∆(Si ) is a k-rationalizable strategy (k ≥ 2) for player i if it is a best response to some strategy profile σ−i ∈ ∆(S−i ) such that each σj is (k − 1)-rationalizable for player j 6= i.

I σi ∈ ∆(Si ) is a rationalizable for player i if it is k-rationalizable for all k ≥ 1.

C. Hurtado (UIUC - Economics) Game Theory 10 / 25 Rationalizability Rationalizability

I Note that the set of rationalizable strategies can no be larger that the set of strategies surviving iterative removal of strictly dominated strategies.

I This follows from the earlier comment that a strictly dominated strategy is never a best response.

I In this sense, rationalizability is (weakly) more restrictive than iterated deletion of strictly dominated strategies.

I It turns out that in 2-player games, the two concepts coincide. In n-player games (n > 2), they don’t have to.

I Strategies that remain after iterative elimination of strategies that are never best responses: those that a rational player can justify, or rationalize, with some reasonable conjecture concerning the behavior of his rivals (reasonable in the sense that his opponents are not presumed to play strategies that are never best responses, etc.).

I ”Rationalizable” intuitively means that there is a plausible explanation that would justify the use of the strategy.

C. Hurtado (UIUC - Economics) Game Theory 11 / 25 Exercises On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Strategy Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory Exercises Exercises

I Exercise 1. Show that there can be no strategy σi ∈ ∆(Si ) such that for all si ∈ Si and s−i ∈ S−i , ui (σi , s−i ) > ui (si , s−i ).

I Exercise 2. Prove that in 2-player games, a pure strategy is never a best response if and only if it is strictly dominated.

I Exercise 3. Determine the set of rationalizable pure strategies for the following game:

1/2 b1 b2 b3 b4

a1 0, 7 2, 5 7, 0 0, 1

a2 5, 2 3, 3 5, 2 0, 1

a3 7, 0 2, 5 0, 7 0, 1

a4 0, 0 0,-2 0, 0 10,-1

C. Hurtado (UIUC - Economics) Game Theory 12 / 25 Pure Strategies Nash Equilibrium On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Strategy Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory Pure Strategies Nash Equilibrium Pure Strategies Nash Equilibrium

I Now we turn to the most well-known in game theory. We’ll first discuss pure strategy Nash equilibrium (PSNE), and then later extend to mixed strategies.

Definition

A strategy profile s = (s1, ..., sN ) ∈ S is a Pure Strategy Nash Equilibrium (PSNE) if for all i and ˜si ∈ Si , u(si , s−i ) ≥ u(˜si , s−i ).

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

I Remark: Every finite game of has a pure strategy Nash equilibrium.

C. Hurtado (UIUC - Economics) Game Theory 13 / 25 Pure Strategies Nash Equilibrium Pure Strategies Nash Equilibrium

I Unlike with our earlier solution concepts (dominance and rationalizability), Nash equilibrium applies to a profile of strategies rather than any individual’s strategy. When people say ”Nash equilibrium strategy”, what they mean is ”a strategy that is part of a Nash equilibrium profile”. I The term equilibrium is used because it connotes that if a player knew that his opponents were playing the prescribed strategies, then she is playing optimally by following her prescribed strategy. I Rationalizability only requires a player play optimally with respect to some ”reasonable” conjecture about the opponents’ play, where ”reasonable” means that the conjectured play of the rivals can also be justified in this way. I Nash equilibrium requires that a player play optimally with respect to what his opponents are actually playing. That is to say, the conjecture she holds about her opponents’ play is correct. I The above point makes clear that Nash equilibrium is not simply a consequence of ( of) rationality and the structure of the game. Clearly, each player’s strategy in a Nash equilibrium profile is rationalizable, but lots of rationalizable profiles are not Nash equilibria.

C. Hurtado (UIUC - Economics) Game Theory 14 / 25 Pure Strategies Nash Equilibrium Pure Strategies Nash Equilibrium

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 99 P sj players. That is, a−i = j6=i 99 . 3 2 I Let the utility of player i be ui (si , s−i ) = 100 − si − 2 a−i . I If each player maximizes his utility, the F.O.C. is:  3  −2 s − a = 0 i 2 −i

3 I Hence, each player i would like to select si = 2 a−i . 3 I Note that a−i ∈ [20, 60], hence 2 a−i ∈ [30, 90]. I Then, si = 20 is dominated by si = 30. That is, regardless of the selection of the others, 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 will select a number between 30 and 60. 3 I With the same logic, a−i ∈ [30, 60], hence 2 a−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.

C. Hurtado (UIUC - Economics) Game Theory 15 / 25 Nash Equilibrium and Strictly Dominated Strategies On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Strategy Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory Nash Equilibrium and Strictly Dominated Strategies Nash Equilibrium and Strictly Dominated Strategies

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

Theorem Every PSNE survives to the Iterated Delation of Strictly Dominated Strategies

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

∗ ui (si , s−i ) > ui (si , s−i ).

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

∗ Hence, si is not a PSNE, which is a contradiction.

C. Hurtado (UIUC - Economics) Game Theory 16 / 25 Nash Equilibrium and Strictly Dominated Strategies Nash Equilibrium and Strictly Dominated Strategies

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

Theorem If the Iterated Delation of Strictly Dominated Strategies reaches a unique solution, this is the unique PSNE.

Proof : Suppose that the unique solution reached by the Iterated Delation of Strictly Dominated Strategies, s∗, is not a PSNE. That means that for some player i there exist si such that ∗ ∗ ∗ ui (si , s−i ) < ui (si , s−i )

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

The uniqueness follows from the previous theorem.

C. Hurtado (UIUC - Economics) Game Theory 17 / 25 Existence of PSNE On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Strategy Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory 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, every point on the straight line segment that joins the pair of points is also within the object. Formally, let V be a vector space over the real numbers. A set S in V is said to be convex if, for all s1 and s2 in S and all µ ∈ [0, 1], the point σ = (1 − µ)s1 + µs2 also belongs to S.

C. Hurtado (UIUC - Economics) Game Theory 18 / 25 Existence of PSNE Existence of PSNE

I Some useful tools: - Extreme value theorem: A real-valued function f is continuous in the closed and 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].

C. Hurtado (UIUC - Economics) Game Theory 19 / 25 Existence of PSNE Existence of PSNE

I Some useful tools:

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

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

Theorem: Suppose f is a continuous real function on a compact metric

space X, let M = supp∈X f (p) and m = infp∈X f (p). Then, exist p and q in X such that f (p) = M and f (q) = m.

proof : Rudin page 89.

C. Hurtado (UIUC - Economics) Game Theory 20 / 25 Existence of PSNE Existence of PSNE

Theorem N (Existence of PSNE). Suppose each Si ⊂ R is non-empty, compact and convex; and each 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 existence Theorem is only useful for infinite games.

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

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

C. Hurtado (UIUC - Economics) Game Theory 21 / 25 Mixed Strategy Nash Equilibrium On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Strategy Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory Mixed Strategy Nash Equilibrium Mixed Strategy Nash Equilibrium

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

Definition

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

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

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

C. Hurtado (UIUC - Economics) Game Theory 22 / 25 Mixed Strategy Nash Equilibrium Mixed Strategy Nash Equilibrium

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 indifferent between playing H or T .

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

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

C. Hurtado (UIUC - Economics) Game Theory 23 / 25 Existence of NE On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Strategy Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory Existence of NE Existence of NE

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

I Proof. For each i, given the finite space of pure strategies, Si , the space of mixed |Si | strategies, ∆(Si ), is a (non-empty) compact and convex subset of R . The utility functions ui : ∆(S) → R defined by " # X X Y ui (σi , σ−i ) = σj (sj ) σi (si )ui (si , s−i )

si ∈Si s−i ∈S−i j6=i

are continuous in σi and quasi-concave in σi . Thus, the previous Theorem implies that there is a pure strategy Nash equilibrium of the infinite normal-form game I, ∆(Si ), ui ; this profile is a (possibly degenerate) mixed strategy Nash equilibrium of the original finite game. I The critical need to allow for mixed strategies is that in finite games, the pure strategy space is not convex, but allowing players to mix over their pure strategies ”convexifies” the space.

C. Hurtado (UIUC - Economics) Game Theory 24 / 25 Exercises On the Agenda

1 Formalizing the Game

2 Rationalizability

3 Exercises

4 Pure Strategies Nash Equilibrium

5 Nash Equilibrium and Strictly Dominated Strategies

6 Existence of PSNE

7 Mixed Strategy Nash Equilibrium

8 Existence of NE

9 Exercises

C. Hurtado (UIUC - Economics) Game Theory Exercises Exercises

I Exercise 4. Find the Nash Equilibria of the following games:

I Exercise 5. Rock (R), Paper (P) or Scissors (S) is a zero sum game. A player who decides to play rock will beat another player who has chosen scissors but will lose to one who has played paper; a play of paper will lose to a play of scissors. a. Find the reduced form of this game is b. Find the Nash Equilibrium of this game. I Exercise 6. Find the PSNE and the MSNE in the following games:

C. Hurtado (UIUC - Economics) Game Theory 25 / 25