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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
Typology: Exercises
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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.]
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?
The Cournot duopoly game is defined by:
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))
]
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)
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.]