IEOR E4407: Game Theoretic Models for Operations
Sample Final (Fall 2011)
1. Esther and Tanya wish to have dinner together. They have two options: dinner at Ollie’s (O)
or at the Terrace (T). Esther first chooses where to go; Tanya observes Esther’s choice and
makes her choice. Esther prefers Ollie’s, while Tanya prefers the terrace. If they eat together,
each gets a payoff of 3 from her preferred choice and 1 from her second choice. (So if they
eat at Ollie’s, Esther’s payoff is 3 whereas Tanya’s is 1) If they eat at different places, each
gets a payoff of zero. All of this information is common knowledge.
(a) List the strategies of Esther and Tanya.
(b) Find a subgame perfect Nash Equilibrium. Also, find a Nash Equilibrium with a different
outcome that is not necessarily subgame perfect.
Consider a following modified game: Tanya does not automatically know Esther’s choice at
no cost. She has an option to observe Esther’s choice or not. Tanya, first chooses between,
“observe” and “do not observe”. If she observes, she learns Esther’s choice and then makes
her own; otherwise she chooses where to go without knowing Esther;s choice. The payoffs are
the same as before.
(c) Draw the extensive form representation of this game.
(d) Find a subgame perfect equilibrium in which both of them eat at the terrace
2. Ross and Daniel simultaneously invest in a project where the level of investment can be any
non-negative real number. If Ross invests xg and Daniel invests xa , Ross and Daniel get a
payoff of tg xg xa −x3g and ta xg xa −x3a respectively. Here, tg and ta are Ross and Daniel’s types.
Each player knows his type, does not know the other’s type but believes that the others’ type
is uniformly distributed between 0 and 1. All of this is common knowledge. Find apsymmetric
Bayesian Nash Equilibrium in which each player’s investment is of the form a + b (t), where
t is the type.
3. Brujis systems is a large software company that makes a popular browser. Each year a new
company emerges that may choose to make a better browser (B) or Not (N). Seeing what this
new company chooses to do, Brujis system may try to improve its browser (I) or Not (N).
The payoffs for (Brujis, Competitor) are: (1,0) for (I,B) ; (3,1) for (I,N); (2,2) for (N,B); and
(4,1) for (N,I). The competitor is out of business no matter what the choices are, and a new
one emerges the following year. Suppose the time horizon is 5 years. The payoff for Brujis
system is its total profit over 5 years; the payoff for each competitor is its annual profit in
the year it lives. The entire history is observable, Find all subgame perfect equilibria.
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4. Consider a third price sealed bid auction. Suppose there are N agents.
(a) Show that the set of strategies in which each player bids her valuation is not a Nash
equilibrium.
(b) Show that for any player i the bid of vi (i.e. truth telling bid) weakly dominates any
other lower bid.
5. Consider an auctioneer with a single good facing 2 risk neutral buyers with independent
valuations uniformly distributed in [0,1] and [1,2] respectively. Suppose the auctioneer uses a
first price auction with a reserve price r. Note that the bidder valuations are not i.i.d unlike
the auction discussed in the class.
(a) What value of r maximizes expected revenue to the seller.
(b) What is the expected revenue to the auctioneer?
6. Consider the first price auction with N bidders and independent private values. Suppose F (.)
is the uniform distribution in [0,1] and the values for the bidders are i.i.d with distribution
F (.)
(a) Given that the object is worth nothing to the seller, and the seller’s reserve price is r,
find the seller’s expected profit as a function of r .
(b) To optimize profit, what value of r should the seller choose?
(c) If the seller uses a posted price mechanism instead of an auction. What is the optimal
posted price p ? (In such a mechanism, a bidder with value greater than p will buy the
object, paying p ) Compare the optimal profit for the seller under this mechanism to the
one you find in part (b) and comment.
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