Econ 414 – Game Theory
Midterm Exam – Fall 2007
Instructor: Stephen Hutton
Time allowed: 75 minutes.
1. (20 points)
Three companies are considering investing in a project. Each firm simultaneously chooses an amount mi to invest in the project. The payoff to firm i is:
(mi – x)(y + zmj+ zmk– mi)
where i, j and k are the indexes for the three firms.
You may assume ∞ > y > x > 1 > z > 0.
a) Set up firm 1’s profit maximization problem. Find firm 1’s best response as a function of m2 and m3. Find best response functions for firms 2 and 3.
Note that this game is identical to the differentiated product Bertrand model we did in class, except for the names of the variables and the fact that we have three firms instead of two.
Assuming an internal solution:
Firm i solves Maxm1(m1 – x)(y + zm2 + zm3 – m1)
This gives FOC: y + zm2 + zm3 – 2m1 + x = 0
m1 = [y + zm2 +zm3 + x]/2
By symmetry, we have:
m2 = [y + zm1 +zm3 + x]/2
m3 = [y + zm2 +zm1 + x]/2
b) Find the unique Nash equilibrium of this game (Hint: show the equilibrium must be symmetric, then impose this to find the solution).
The solution exists when we solve the best responses simultaneously:
m1 = [y + zm2 +zm3 + x]/2
m2 = [y + zm1 +zm3 + x]/2
m3 = [y + zm2 +zm1 + x]/2
Note that the first two equations imply:
m1 – m2 = [y + zm2 +zm3 + x]/2 - [y + zm1 +zm3 + x]/2
m1 – m2 = zm2/2 - zm1/2
= (m2 – m1)z/2
which implies m1 = m2.
By comparing the first and last equations, we similarly get m1 = m3.
So we have m1 = m2 = m3 = m*. Ie the solution is symmetric. Imposing this on any of the best responses we have m* = [y + zm* +zm* + x]/2 = (y + x)/2 + zm*
m*(1 – z) = (x + y)/2
m* = (x+y)/[2(1-z)]
Thus, the unique equilibrium is for each firm to invest m = (x+y)/[2(1-z)]
2. Mixed strategy equilibria (30 points)
Find all Nash equilibria, in both pure and mixed strategies, for the following game. Are any strategies strictly dominated by a pure strategy?
Player 2L / M / R
T / 1,3 / 2,4 / 3,2
Player 1 / M / 1,2 / 2,1 / 4,1
B / 2,4 / 1,4 / 2,2
First, note that R is strictly dominated by L, so we can eliminate R. We need not consider any cases where R is played with any positive probability.
Two pure strategy Nash equilibria, (T,M) and (B,L).
Partial mixing strats:
Nothing where player 1 plays T or M and 2 mixes.
(0,0,1) (q, 1-q, 0)
2 is willing to mix for any q.
For player 1 to play B, we need 2q + (1 – q) ≥q + 2(1 – q)
Ie q ≥½
So, we have a set of equilibria (0,0,1) (q, 1-q, 0) for q ≥½.
Nothing where 2 plays L and 1 mixes between B and something else.
Nothing where 2 plays M and 1 mixes between B and something else.
(p, 1-p, 0),(1,0, 0)
1 is willing to mix for any p.
For player 2 to play L, we need 3p + 2(1-p) ≥4p + (1 – p)
Ie 1 – p ≥p
p ≤½.
So we have a set of equilibria (p, 1-p, 0),(1,0, 0) for p ≤½.
(p, 1-p, 0), (0,1, 0)
1 is willing to mix for any p.
For player 2 to play M, we need 3p + 2(1-p) ≤4p + (1 – p)
p ≥½.
So we have a set of equilibria (p, 1-p, 0),(0,1, 0) for p ≥½.
(p, 1-p, 0), (q, 1-q, 0)
Player 1 gets equal payoffs from T and M for any q.
Player 2 gets equal payoffs from L and M when 3p + 2(1-p) =4p + (1 – p)
Ie when p = ½ .
So we have a set of equilibria (1/2, 1/2, 0), (q, 1-q, 0) for any q.
(p, 0, 1-p), (q, 1-q, 0)
q + 2(1-q) = 2q + (1 – q)
Ie: q = ½ .
3p + 4(1-p) = 4p + 4(1-p)
Implies p = 0
We have already done this case.
(0, p, 1-p), (q, 1-q, 0)
q + 2(1-q) = 2q + (1 – q)
Ie: q = ½ .
2p + 4(1-p) = p + 4(1-p)
implies p = 0.
We have already done this case.
(p, r, 1-p-r), (q, 1-q, 0)
q + 2(1-q) = 2q + (1 – q)
Ie: q = ½ .
3p + 2r + 4(1 – p – r) = 4p +r + 4(1 – p – r)
r = p
So we have a set of equilibria (p, p, 1-2p), (1/2, 1/2, 0) for any p < ½.
3. Voluntary pricing (25 points)
A band called StereoArm has come up with a new business model for releasing their new album over the internet. Rather than charging customers a fixed price per download, they let each customer i choose their own price pi to pay for the album, and pi may be any non-negative number, including zero.
Suppose that there are n identical consumers (indexed by i = 1,2,…,n) all of whom value the album at $10. The n consumers all simultaneously choose their price pi, and receive a payoff of 10 – pi.
a)Do any players have a dominant strategy in this game? What is the unique Nash equilibrium of this game?
First, note that the band is not a player in this game, either in this question or the questions to follow, because their actions are completely dictated. The only players are the n consumers.
All n players have a dominant strategy to choose a price pi = 0. The unique Nash equilibrium is for all players to choose this price.
Now suppose that the band is considering making a second album. Initially,the n consumers simultaneously choose what prices p1i they will pay for the first album. If total revenue (ie the sum of all p1i’s) from the first album is less than $1000 then the game ends. If the total revenue is at least $1,000, then the band releases a second album, with the same price system as the first album.
If a second album is produced, then the n consumers will simultaneously choose what prices p2i they will pay for the second album, and then the game ends.
Consumers value both albums at $10 each, and their payoff is 10 for each album produced minus the prices they pay for each album. So if two albums are produced, then consumer i will receive 20 – p1i – p2i.
b)Suppose n = 100. Give an example of a subgame perfect Nash equilibrium where the band produces two albums, and explain why this is an equilibrium. If n < 100 can such an equilibrium exist? Why or why not? If n > 100 can such an equilibrium exist? Why or why not?
As long as there are at least 100 consumers, there exists an equilibrium where a second album is produced. If 100 consumers pay p1i = 10 and p2i = 0 (and all other customers pay p1i = 10 and p2i = 0) then we have a subgame perfect Nash equilibrium. Each player receives a payoff of 10 in this equilibrium.
No player can profitably deviate; any player who increases their price just lowers their payoff, and any player who reduces their p1i cancels the second album and so gets at most a payoff of 10 (if they reduce theirp1i to zero).
If fewer than 100 consumers existed, then in order for the second album to be produced at least one player would have to pay p1i > 10, which would lead to a total payoff < 10 for that player, and so that player could profitably deviate to p1i = 0 and get a total payoff ≥10. Hence there is no equilibrium where both albums are produced for n < 100.
c)Fully describe the set of all subgame perfect Nash equilibria to this game, for any value(s) of n.
(Note: if you cannot fully describe the set, at least describe properties that must be true in any equilibrium.)
[First, note that it is NOT necessary for all consumers to pay the same price in an equilibrium. Symmetric game setup and symmetric best response functions DO NOT guarantee that all equilibria have all players take the same action. It does guarantee that for any equilibrium where players take different actions, there will also be another similar equilibrium where different players take the same roles as do players in the initial equilibrium. Eg if there is a game that is fully symmetric between player 1 and player 2, there could be an equilibrium where 1 takes action X and 2 takes action Y. But then there must also be an equilibrium where 1 takes action Y and 2 takes action X.]
In all subgame perfect Nash equilibria, all consumers will choose p2i = 0, as this strictly dominates any other choice.
In any equilibrium, total first album revenue will never exceed $1000, otherwise any player has an incentive to profitably deviate by lowering their choice of p1i, which increases their profit without changing the availability of a second round album.
If total first album revenue is less than $1000, then no consumer will ever choose any p1i > 0, as they could profitably deviate to p1i = 0.
No player will ever choose any p1i> 10. If they did so, they would receive a total payoff < 10, so they could profitably deviate to choosing p1i = 0.
If n < 100, the unique equilibrium is for p1i = p2i = 0 for all i.
If n ≥100, then there are two types of equilibria; one type where the second album is produced, and one where it is not.
If the second album is not produced, then we will have p1i = p2i = 0 for all i.
If the second album is produced, then we will have a set of equilibria where a set of k consumers (k is a weak subset of n, and has at least 100 members) choose p1i such that:
1) Σk p1i = 1000,
2) 0 < p1i≤10 for all consumers in k,
3) Any consumers not in k choose p1i = 0.
4) All consumers choose p2i = 0.
This is an equilibrium because all consumers in k receive payoffs ≥10, but if any of them deviate by reducing their price, the second album will no longer be produced, so they will receive a payoff ≤10. Thus, no consumer in k has any profitable deviation. And clearly no other player can profitably deviate.
Example:
n = 400
Players 1, 2….100 choose p1i = 3, p2i = 0,
Players 101, 102….200 choose p1i = 2, p2i = 0,
Players 201, 202….300 choose p1i = 5, p2i = 0,
Players 301, 3022….400 choose p1i = 0, p2i = 0.