ECON0027 Game Theory
Home assignment 1
1. In his book Leo the African, Amin Malouf relates the following story:
Halfway between Fez and Meknes we stopped for the night in a village called ‘Ar, Shame. The imam of the mosque ofered to accomodate us in return for
a donation for the orphans whom he looked after. He... lost no time in telling us why this village should have such a name. The inhabitants, he informed us,
had always been known for their greed and used to sufer from this reputation. The merchant caravans avoided it and would not stop there. One day, having
learned that the King of Fez was hunting lions in the neighborhood, they decided to invite him and his court, and killed a number of sheep in his honour. The
sovereign had dinner and went to bed. Wishing to show their genoros- ity, they placed a huge goatskin bottle before his door and agreed to ill it up with milk
for the royal breakfast. The villagers all had to milk their goats and then each of them had to go to tip his bucket into the container. Given its great size, each
of them said to himself that he might just as well dilute his milk with a good quantity of water without anyone noticing. To the extent that in the morning
such a thin liquid was poured out for the king and his court that it had no other taste than the taste of meanness and greed.
(a) Set out the interaction between the villagers as a game in strategic form, spec- ifying clearly how the payofs to each villager depends upon their individual actions and those of others.
(b) Analyze the equilibria of the game you have constructed, explaining whether the situation that arose was unavoidable, or could have been prevented.
(c) Deine a strictly dominated strategy. Suppose that strategy si is strictly dom- inated for player i. Show that si cannot be played in a pure strategy Nash equilibrium. Show also that it cannot be played with positive probability in a mixed strategy Nash equilibrium.
2. Consider a game of n players in which each player chooses an efort level ei ≥ 0, i = 1, .., n. The marginal beneit of efort for player i depends on the efort exerted by the other players. In particular, the payof of player i is
where parameters ai > 0, wij ≥ 0 are commonly known for all i and j. The players choose their efort levels simultaneously and independently.
(a) Solve for a best reply of player i.
(b) Assume that ai = a > 0 and wij = w for all i,j. Provide the conditions under which symmetric NE in pure strategies exists. Provide intuition for your result.
(c) Assume that n = 3, ai = a > 0. Consider two situations: irst w12 = w23 = w13 = w, and second w12 = w23 = w, w13 = 0. Assuming that 0 < w < 1/2, compare the equilibrium efort level of player 2 in these two situations (you may impose reasonable symmetry assumptions on equilibrium). Provide intuition for your result.
3. Consider the game depicted below.
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L
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R
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T
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a11 , b11
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a12 , b12
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B
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a21 , b21
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a22 , b22
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(a) Deine a weakly dominated strategy and set out conditions on the payofs so that player 1 (the row player) has a weakly dominated strategy.
(b) Set out conditions on the payofs such that the game has two pure strategy equilibria, (T, L) and (B, R).
(c) A Nash equilibrium (s1 , s2 ) is said to be strict if s1 is the unique best response to s2 and s2 is the unique best response to s1 (i.e. each player incurs a payof loss by deviating from his equilibrium strategy). Assume that (T, L) and (B, R) are both strict Nash equilibria, and solve for a completely mixed Nash equilibrium, i.e. one where each player randomizes between his two pure strategies, as a function of the payofs, aij and bij . Show that if the Nash equilibria (T, L) and (B, R) are not strict, then there may not be a completely mixed Nash equilibrium.
4. Consider a road which is represented by the interval [0, 1]. Let a be a number such that 0 < a < 1. Vendor 1 can locate at any point on the interval [0, a] (that is, he can locate at any point x such that 0 ≤ x ≤ a). Vendor 2 can locate at any point on the interval [a, 1]. A unit mass of onsumers are uniformly distributed on [0, 1] and each consumer buys one unit of the good from the vendor who is closest to him. If the two vendors locate at the same point a, then each gets one-half of the consumers.
The game is as follows. Vendors choose locations simultaneiously, and a vendor’s payof is given by the number of consumers who purchase from him.
(a) Write down the strategy sets and payof functions in this game.
(b) Suppose a = 0.5. Show that this game has a unique Nash equilibrium in pure strategies. That is, you need to show (i) there is a Nash equilibrium, and (ii) there is no other Nash equilibrium.
(c) Suppose a < 0.5. Show that the game does not have a Nash equilibrium in pure strategies.