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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.]
