14.12 Game Theory
Muhamet Yildiz
Fall 2012
Homework 1
Due on 9/25/2012
1. Consider a homeowner with Von-Neumann and Morgenstern utility function u, where
u (x) = 1 − e−x for wealth level x, measured in million US dollars. His entire wealth
is his house. The value of a house is 1 (million US dollars), but the house can be
destroyed by a fiood, reducing its value to 0, with probability π ∈ (0, 1).
(a) What is the largest premium P is the homeowner is willing to pay for a full
insurance? (He pays the premium P and gets back 1 in case of a fiood, making
his wealth 1 − P regardless of the fiood.)
(b) Suppose there is a local insurance company who has insured n houses, all in
his neighborhood, for premium P. Suppose also that with probability π there
can be fiood in the neighborhood destroying all houses (i.e., either all houses are
destroyed or none of them is destroyed). Suppose finally that P is small enough
that the homeowner has insured is house. Having insured his house, what is the
largest Q that he is willing to pay to get the 1/n share of the company? (The
value of the company is the total premium it collects minus the payments to the
insured homeowners in case of a fiood.)
(c) Answer part (b) assuming now that the insurance company is global. It insured n
houses in different parts of the world (all outside of his neighborhood), so that the
destruction of houses by fiood are all independent (i.e., the probability of fiood in
one house is π independent of how many other houses has been fiooded).
n k/nπk (1 − π)
n−k ∼ π+π(1−π)/(2n) (d) Assume that n is large enough so that k=0 Cn,ke = e ,
discuss your answers to above questions (briefiy). [Here, Cn,k denotes the number
of k combinations out of n, and the sum is one minus the expected payoff from
the loss due to the payments to the fiooded houses.]
2. Consider the game in which the following are commonly known. First, Ann chooses
between actions a and b. Then, with probability 1/3, Bob observes which action Ann
has chosen and with probability 2/3 he does not observe the action she has chosen. In
all cases (regardless of whether he has observed Ann chose a, or he has observed Ann
chose b, or he has not observed any action), Bob chooses between actions α and β.
The payoff of each player is 1 after (a, α) and (b, β) and 0 otherwise.
(a) Write the above game in extensive form.
(b) Write the above game in normal form.
3. Consider the following variation of the above game. First, Ann chooses between actions
a and b. Then, Bob decides whether to observe the chosen action of Ann or not, by
choosing between the actions Open and Shut, respectively. In all cases, Bob then
1
chooses between actions α and β. The payoff of Ann is 1 after (a, α) and (b, β) and
0 otherwise, regardless of whether Bob chooses Open or Shut. The payoff of Bob is
equal to the payoff of Ann if he has chosen Shut, and his payoff is equal to the payoff
of Ann minus 1/2 if he has chosen Open.
(a) Write the above game in extensive form.
(b) Write the above game in normal form.
4. Federal government is planning to build an interstate highway between two states,
named A and B. The highway costs C > 0 to the government, and the value of the
highway to the states A and B are vA ≥ 0 and vB ≥ 0, respectively. Simultaneously,
each state i ∈ {A, B} is to bid bi ∈ [0, ∞). If bA + bB ≥ C the highway is constructed.
For any distinct i, j ∈ {A, B}, state i pays C − bj to the federal government if bj <
C ≤ bA + bB. (There is no payment otherwise.) The payoff of a state is the value of
the highway to the state minus its own payment to the government if the highway is
built, and 0 otherwise. (You can focus on the case vA + vB < C.)
(a) Write this in the normal form.
(b) Check if there is a dominant strategy equilibrium, and compute it if there is one
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