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Nash Equilibria in Game Theory: Proofs and Exercises, Exercises of Game Theory

Exercises on nash equilibria in game theory, including proofs for statements relating to dominance solvability and best responses. Topics covered include the stag hunt game, cournot duopoly, and matching pennies. Students will learn how to compute players' payoffs, find best responses, and draw best response correspondences.

What you will learn

  • How do you find a player's best response in a game?
  • What is the relationship between Nash equilibria and dominance solvability?
  • What is the significance of the intersection of best response correspondences in a game?

Typology: Exercises

2021/2022

Uploaded on 02/11/2022

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Week 4. Static games with complete information III: Nash
equilibria
Exercise 1: Nash equilibrium vs dominance solvability
Prove the following statements:
(i) If a pure strategy s(i)
jis dominated by a pure strategy s(i)
kand σ= (σ(1), . . . , σ(n)) is a Nash equilibrium,
then σ(i)
j= 0.
(ii) If the game is dominance solvable such that the unique outcome of iterated elimination of dominated
strategies is some pure strategy s= (s(1), . . . , s(n)), then sis a Nash equilibrium.
[Suggestion: One could use contradiction to prove the above statements. For example, for (i) assume that
these was a Nash equilibrium with σ(i)
j>0, and show that this would yield some contradiction.]
Exercise 2: Best responses
Consider the stag hunt game:
player 2
Stag Hare
player 1 Stag (10,10) (0,6)
Hare (6,0) (6,6)
Suppose player 1 uses the mixed strategy (x, 1x), where xis player 1’s probability to Stag. Similarly,
player 2’s strategy is (y, 1y).
(i) For given x, y compute the players’ payoffs π(1)(x, y), π(2) (x, y) (see Remarks 2.6, 2.7).
(ii) For a given ycompute player 1’s best response (BR(y)). In particular, show that there is some ysuch
that all x[0,1] are a best response.
(iii) Draw the two best response correspondences BR(x), BR(y) into a xyplane. How often do they
intersect? What does it mean if they intersect?
1
pf3

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Week 4. Static games with complete information III: Nash

equilibria

Exercise 1: Nash equilibrium vs dominance solvability

Prove the following statements:

(i) If a pure strategy s( ji )is dominated by a pure strategy s( ki )and σ = (σ(1),... , σ(n)) is a Nash equilibrium, then σ( ji )= 0.

(ii) If the game is dominance solvable such that the unique outcome of iterated elimination of dominated strategies is some pure strategy s = (s(1),... , s(n)), then s is a Nash equilibrium.

[Suggestion: One could use contradiction to prove the above statements. For example, for (i) assume that

these was a Nash equilibrium with σ (i) j >^ 0, and show that this would yield some contradiction.]

Exercise 2: Best responses

Consider the stag hunt game:

player 2

( Stag^ Hare) player 1 Stag (10, 10) (0, 6) Hare (6, 0) (6, 6)

Suppose player 1 uses the mixed strategy (x, 1 − x), where x is player 1’s probability to Stag. Similarly, player 2’s strategy is (y, 1 − y).

(i) For given x, y compute the players’ payoffs π(1)(x, y), π(2)(x, y) (see Remarks 2.6, 2.7).

(ii) For a given y compute player 1’s best response (BR(y)). In particular, show that there is some y∗^ such that all x ∈ [0, 1] are a best response.

(iii) Draw the two best response correspondences BR(x), BR(y) into a x − y plane. How often do they intersect? What does it mean if they intersect?

Exercise 3: Cournot Duopoly

The Cournot duopoly game is defined by:

  • Players: N = {Firm 1, Firm 2}
  • Actions: Amount of good produced, x(i)^ ∈ [0, ∞) for i ∈ { 1 , 2 }
  • Payoffs: π(i)(x(1), x(2)) = [a − b(x(1)^ + x(2))]x(i)^ − cx(i)

Show that there is a Nash equilibrium in pure strategies. For simplicity assume a = 10, b = 1 , c = 1.

[Hint: For each x(i)^ computer BR(x(−i)). Then solve simultaneously:

x(1)^ = BR(x(2)) x(2)^ = BR(x(1))

]

Exercise 4: Matching Pennies

Compute the Nash equilibria for the following two games, and interpret the result.

( Left^ Right ) Top (0. 8 , 0 .4) (0. 4 , 0 .8) Bottom (0. 4 , 0 .8) (0. 8 , 0 .4)

( Left^ Right ) Top (3. 2 , 0 .4) (0. 4 , 0 .8) Bottom (0. 4 , 0 .8) (0. 8 , 0 .4)

Bonus Exercise 1: Verifying NE in games with finitely many players & actions

Show that to verify whether a strategy profile ˆσ = (ˆσ(1),... , σˆ(n)) is a Nash equilibrium, it is sufficient to check all deviations towards pure strategies.

Specifically show that σˆ is a Nash equilibrium if and only if for all players i the following two conditions hold:

(i) All actions that player i uses give the same payoff: if σ (i) j >^ 0 and^ σ

(i) k >^ 0 then^ π

(i)(s(i) j ,^ σˆ (−i)) =

π(i)(s( ki ), σˆ(−i)).

(ii) Actions that are not played are not profitable: if σ (i) j = 0 then^ π (i)(s(i) j ,^ σˆ (−i)) ≤ π(i)(ˆσ(i), ˆσ(−i)).

[Hint: One way to prove the above is once again by contradiction.]