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# ps4 - Ec1052 Introduction to Game Theory Harvard University...

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Ec1052: Introduction to Game Theory Handout 10 Harvard University 23 March 2004 Problem Set 4 Due: Saturday, 10 April, 5pm sharp (Irit’s folder at Littauer) Challenging problems are marked with one star. Double-starred questions do NOT count towards the grade. They are very hard and simply for the intellec- tually curious. Remember that you can work in groups but you have to write up your own solutions. Good luck! Problem 1. Is the concept of Nash equilibrium too weak or too strong a solution concept? Problem 2. Show that each of the games below has no ESS. B A A B 0,0 0,0 0,0 0,0 C B A A B C 2,2 0,3 3,0 3,0 2,2 0,3 0,3 3,0 2,2 Problem 3. Find all the steady states of the second game (above) under the replicator dynamics. 3(a) ** Show that none of them is asymptotically stable.

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Ec1052 Handout 10: Problem Set 4 2 Problem 4. Let G be a finite symmetric game. 4(a) Show that x is an ESS if and only if for all y 6 = x either u ( x, x ) > u ( y, x ) or u ( x, x ) = u ( y, x ) and u ( x, y ) > u ( y, y ). 4(b) Show that x is an ESS implies that ( x, x ) is a Nash equilibrium but that the converse is not true. 4(c) Show that any symmetric strict Nash equilibrium is an ESS. 4(d) * Show that if x and y are distinct ESS then they cannot have the same support. What does this imply about the maximum number of ESSs G can have? Problem 5. * Show that fictitious play respects IDSDS. Formally, show that the assessment of player i , γ t i ( s - i ) 0 as t → ∞ for s - i 6∈ S - i .
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