Game Theory and Strategic Decisions
Problem Set # 2
Istanbul Harvard Summer School Program, 2009
due Tuesday, July 7 (at the beginning of class)
1. A dispute between the NBA and the players union (5 pts.)
Read the column about the labor negotiations during the 1998-99 National Basketball Association
Lockout by Ira Berkow (New York Times, January 6, 1999).
(Available at: http://query.nytimes.com/gst/fullpage.html?res=9A0DE2DB153EF935A35752C0A96F958260 )
Recall that Berkow begins his column with the story about Uncle Sile and Uncle Howard.
(a) In one paragraph, explain why Berkow would use this story as an analogy for labor-management
At times situations arise in negotiations where a negative downward spiral threatens to ―take the
participants over a cliff.‖ Such mutually-assured-destruction is appealing to participants in the
heat of the moment but afterwards often looks foolhardy. Recognizing the game being played
between the owners and players as mutually destructive, Berkow confronts the two sides with the
negative possible outcomes of their behavior and forces them to reconsider their strategies in
light of the potential outcomes and incentives that he presents. By forcing the players to step
back and look at the whole game it is sometimes possible to avert a disaster.
This is a chicken game, in which both sides have incentive to appear tough; if they
both act tough, however, a mutually destructive outcome occurs.
2. Nash Equilibrium (5 pts.)
“Since action B leads to a mutually worse result, the outcome in which both players pick action B
cannot possibly satisfy the definition of Nash Equilibrium.” True or false? Explain.
False. Once again look at the prisoner’s dilemma—by having each player choose
Confess the game moves to a Nash Equilibrium that is mutually worse than the result
if each player chose Don’t Confess.
3. Lending poor in developing countries (20 pts.)
Consider a developing country in which a group of poor borrowers and lenders play the following
“trust” game. $100 loan is able to generate a 100 percent return on investment. Assume that the interest
rate is 30 percent, so that $130 is due back to the Lender at payback time. The normal form of the game
(presented for one borrower and one lender) is as follows.
Repay Loan Don‟t Repay
Lend 30, 70 -100 , 200
Don‟t Lend 0,0 0,0
(a) What is the Nash Equilibrium of this game?
The Nash Equilibrium is (Don’t Lend, Don’t Repay).
(b) Assume that you are a policy maker, and you aim at achieving a better outcome than the Nash
equilibrium outcome in this game. Assume that you can modify one payoff in one of the four outcomes
for one (and only one) of the players in this game.
Which payoff would you modify, and how (assign a number for the payoff that replaces the current
one)? Can you think of a policy tool that would indeed yield the change you have suggested?
The best option is to change the payoff to the Borrower for the strategy-box (Lend, Don’t Repay)
such that the borrower receives less than 70 for not repaying (instead of 200).
By decreasing the benefit for ―Don’t Repay‖ to the Borrower, such that the borrower gets less
than 70 from cheating and not repaying, the lender is ensured that the borrower has incentive to
repay the loan (so that s/he can get the full 70) and thereby an equilibrium of (Lend, Repay
Loan) can be achieved.
This policy change could be achieved in many ways, for example by strengthening the
courts such that lenders can seize collateral in the event of non-payment more easily.
(c) What is the Nash equilibrium of the game with the new payoff you have assigned?
There are now two Nash Equilibria: (Don’t Lend, Don’t Repay) and the new Nash Equilibrium
is (Lend, Repay Loan).
4. It takes two to play tennis, but how much dope to win? (15 pts.)
There are two tennis players A and B who play against each other in a tournament. The winner will
reach the next round. Both of the players have two strategies; they can either take doping (D) or they can
choose not to take doping (N). Both are equally good players, meaning they win with 50 percent
probability if both take doping or if both don‟t take doping. If only one of the two takes doping, he or
she will win for sure. Winning gives a pleasure (utility) of 2 units, losing gives displeasure of 2 units.
Additionally, the doping costs 1 unit.
(a) Present this game in normal form.
D -1,-1 1,-2
N -2,1 0,0
Note: EV (Doping)= 0.5*2+0.5*(-2) + (-1)= -1
(b) Find all Nash equilibria.
(c) What would you do as a policy maker if you want to regulate doping given the incentives in the
„doping game‟? (This can involve actions that change the outcomes of the game…)
I would have to increase the penalty for doping such that whenever a player takes doping the
payoff is worse than when they do not take doping (and thus also the payoff when they both take
doping would be worse still). For example increase the cost of doping to 3 units by raising the
penalty to those caught doping.
5. Mixed Strategies in Battle (25 pts.)
Two nations, Tankistan and Weaponland, are at war. A Tankistan unit is lost in the jungle along the
border of the two countries, and the leader of the unit notices one night that a Weaponland unit is hot on
their trail. Indeed, based on their maneuvering, it looks like they might attack the next day. The
Tankistan unit has two options – to either prepare a surprise early-morning counter-attack (which would
be great if the Weaponland unit is sleeping in to prepare for an afternoon attack), or to stay up all night
fortifying their location to prepare for a morning siege (which would be great for a morning attack but
would leave the unit tired in the event of a afternoon attack).
The payoffs are represented in the following table:
Attack in AM Attack in PM
0, 100 300, -200
200, 0 -100, 300
Siege in AM
(a) Are there pure strategy Nash Equilibria in this game? If so, what are they?
No, using best responses we find that there is no pure strategy equilibrium (see
underlines in the table above).
(b) What is the mixed-strategy equilibrium of the above game?
First, we assign probabilities to all possible pure strategies for the players:
Tankistan, Early AM Counter-Attack: probability = p
Tankistan, Prepare for AM Siege: probability = (1-p)
Weaponland, Attack in AM: probability = q
Weaponland, Attack in PM: probability = (1-q)
Tankistan wants to make Weaponland indifferent between its two pure strategies, AM
and PM (see below for algebra):
EVW(AM) = 100p + 0(1-p)
EVW(PM) = -200p+300(1-p) = -500p+300
Set these two expected values equal:
100p = -500p+300 p = 1/2
Therefore at equilibrium, Tankistan will choose to play ―Early AM Counter Attack‖
with 50% probability and ―Prepare for AM Siege‖ with 50% probability.
Weaponland wants to make Tankistan indifferent between its two pure strategies,
Counter and SiegePrep (see below for algebra):
EVT(Counter) = 0(q) + 300(1-q)
EVT(SiegePrep) = 200q + (-100)(1-q)
Set these two expected values equal:
300-300q = 300q – 100 q = 2/3
Therefore at equilibrium, Weaponland will play ―AM‖ with 66.67% probability and
―PM‖ with 33.33% probability.
(c) In the equilibrium in (b), which unit is better off, Tankistan or Weaponland? Assume here that the
payoffs in the boxes represent “satisfaction points” that each unit receives in the aftermath of battle and
that they are comparable.
Using the probabilities from part (b), we know:
Tankistan gets: 0 with 1/3 probability; 300 with 1/6 probability; 200 with 1/3
probability; and -100 with 1/6 probability Expected Value of Game = 100
Weaponland gets: 100 with 1/3 probability; 0 with 1/3 probability; -200 with 1/6
probability; 300 with 1/6 probability Expected Value of Game = 50
So Tankistan is better off that Weaponland at the Mixed Equilibrium.
6. Minimax in Soccer (15 pts.)
Two big soccer clubs, Manchester United and Chelsea, are preparing for a game. Each team can take
one of three strategies in the upcoming game – a defensive strategy, a standard strategy, or an attacking
strategy. The payoffs for Manchester United in each of the potential strategy combinations are listed
below (this is a zero-sum game, so Chelsea‟s payoffs are equal to [–(Manchester United payoff)] for
Defensive Standard Attacking
Defensive -2 1 1
Standard 0 -2 -5
Attacking 1 5 3
(a) Solve for the Nash equilibrium of this game using a minimax/maximin approach from Manchester
United‟s perspective. That is, determine which strategy is a maximin for Manchester United and which
Chelsea strategy results in a minimax payoff for Manchester United.
We want to find the pure strategy that maximizes the minimum payoff for Manchester United in
this game (the ―maximin‖). That strategy is ―Attacking‖ (min=1 for this strategy, which is
greater than -2 for Defensive and -5 for Standard).
We also want to find the Chelsea strategy that minimizes the maximum possible payoff for
Manchester United (the ―minimax‖). This strategy is ―Defensive‖ (max for United with this
Chelsea strategy is 1, which is less than 5 for Standard and 3 for Attacking).
Since both the maximin and the minimax lead us specifically to the payoff the bottom left box in
the game (Attacking, Defensive), this is the Nash Equilibrium of the game.
(b) Now re-draw the table with Chelsea‟s payoff included, and solve for the Nash equilibrium using best
responses or iterated elimination of dominated strategies. [Hint: your answers to (a) and (b) should be
With Best Responses (see table below)
Defensive Standard Attacking
Defensive -2, 2 1, -1 1, -1
Standard 0, 0 -2, 2 -5, 5
Attacking 1, -1 5, -5 3, -3
This confirms the (Attacking, Defensive) Nash equilibrium.
With IESD: Attacking strictly dominates both Defensive and Standard for Manchester United
(indeed, it is a dominant strategy!). We can then eliminate these two row options for United.
That leaves only Chelsea able to choose a strategy, and it will choose Defensive, which is now its
dominant strategy in the reduced, one-row game! Again, the Nash equilibrium is the same as in
7. Finding Nash Equilibria (15 pts.)
In the following games, find all Nash Equilibria in pure strategies, if any (note that these games are
identical to those in Problem Set #1, question 3).
Left Middle Right
Up 0,1 9,0 2,3
Player 1 Straight 5,9 7,3 1,7
Down 7,5 10 , 10 3,5
The pure strategy N.E. is (Down, Middle).
West Center East
North 2,3 8,2 10 , 6
Straight Up 3,0 4,5 6,4
Down 5,4 6,1 2,5
South 4,5 2,3 5,2
The pure strategy N.E. is (North, East).
A 5,5 0,6
Player 1 B 8,4 3,1
C 4,5 5,3
The pure strategy N.E. is (B, A).