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Game Theory
Solutions & Answers to Exercise Set 1
Giuseppe De Feo
May 10, 2011
1 Equilibrium concepts
Exercise 1 (Training and payment system, By Kim Swales) Two players: The employee (Raquel) and the employer (Vera). Raquel has to choose whether to pursue training that costs £ 1 , 000 to herself or not. Vera has to decide whether to pay a fixed wage of £ 10 , 000 to Raquel or share the revenues of the enterprise 50:50 with Raquel. The output is positively affected by both training and revenue sharing. Indeed, with no training and a fixed wage total output is £ 20 , 000 , while if either training or profit sharing is implemented the output rises to £ 22 , 000. If both training and revenue sharing are implemented the output is £ 25 , 000.
- Construct the pay-off matrix
- Is there any equilibrium in dominant strategies?
- Can you find the solution of the game with Iterated Elimination of Dominated Strategies?
- Is there any Nash equilibrium?
Solution.
- This game has the following characteristics:
- Players: Raquel and Vera
- Strategies:
- Raquel’s: pursue training (costly to herself: £ 1 , 000), or not
- Vera’s: give revenue sharing (50:50), or fixed wage (£10,000)
- payoffs: depend on total output and the way it is split. Output depends positively upon two factors: whether Raquel has training and if Vera adopts profit sharing. - Fixed wages + no training: output = 20,
- Add either training or revenue share: output = 22,
- Both training and revenue share: output = 25, We can then build the payoff matrix: with unit of account: £′ 000
Raquel
Vera Revenue sharing Fixed wage Training 11. 5 , 12. 5 9 , 12 No training 11 , 11 10 , 10
- No, there is no equilibrium in dominant strategies because Raquel has no dominant strategy. She prefers to train only if Vera gives revenue sharing, while prefers not to train with a fixed wage.
- Yes. Fixed wage is a dominated strategy for Vera. Assuming that players are rational and that this information is common knowledge, Raquel knows that Vera will never choose a fixed wage. Then she will choose to train because No training is a dominated strategy after the elimination of Vera’s dominated strategy.
- Yes. Every equilibrium identified by Iterated Elimination of Dominated Strategies is a Nash equilibrium.
Exercise 2 (Simultaneous-move games) Construct the reaction functions and find the Nash equilibrium in the following normal form games.
Will and John 1
John
Will Left Right Up 9 , 20 90 , 0 Middle 12 , 14 40 , 13 Down 14 , 0 17 , − 2
Will and John 2
John
Will Left Centre Right Up 2 , 8 0 , 9 4 , 3 Down 3 , 7 − 2 , 10 2 , 15
Will and John 3
John Will Down Left John’s R.F. (^) Middle Right
Will John Left Up Will’s R.F. Right Middle Right Down
The Nash equilibrium is defined by mutually consistent best responses: therefore {middle, right} is the unique Nash equilibrium of the game.
Exercise 3 (by Kim Swales) The table below represents the pay-offs in a one-shot, simultaneous move game with com- plete information. (Player As pay-offs are given first)
Player A
Player B Left Middle Right Top 7 , 17 21 , 21 14 , 11 Middle 10 , 5 14 , 4 4 , 3 Bottom 4 , 4 7 , 3 10 , 25
- Find the Nash equilibria in pure strategies for the game whose pay-offs are represented in the table above.
- What is the likely focal equilibrium and why?
Exercise 4 (by Kim Swales) Companies A and B can compete on advertising or R+D. The table below represents the pay-offs measured in profits (£, million) in a one-shot simultaneous move game, with complete information. Company A’s profits are shown first.
Company A
Company B Advertising R&D Advertising 50 , 25 10 , 70 R&D 20 , 40 60 , 35
- Find the mixed strategy equilibrium.
- What are the expected pay-offs for both firms?
2 Prisoners’ Dilemma games
Exercise 5 (A prisoner’s dilemma game, by Kim Swales) Firms Alpha and Beta serve the same market. They have constant average costs of £ 2 per unit. The firms can choose either a high price (£10) or a low price (£5) for their output. When both firms set a high price, total demand = 10,000 units which is split evenly between the two firms. When both set a low price, total demand is 18,000, which is again split evenly. If one firm sets a low price and the second a high price, the low priced firm sells 15,000 units, the high priced firm only 2,000 units.
Analyse the pricing decisions of the two firms as a non-co-operative game.
- In the normal from representation, construct the pay-off matrix, where the elements of each cell of the matrix are the two firms’ profits.
- Derive the equilibrium set of strategies.
- Explain why this is an example of the prisoners’ dilemma game. Solution.
- The pay-off for firm i = α, β is total profits, Πi, which equals total revenue, T Ri minus total cost, T Ci. Therefore, for the following sets of strategies: (a) {High price, High price}. Total demand is equal to 10, 000 and so each firm sells 5 , 000 units. T Ri = 5 , 000 × 10 = 50, 000 T Ci = 5 , 000 × 2 = 10, 000 Πi = 50, 000 − 10 , 000 = 40, 000 ∀i = α, β (b) {Low price, Low price}. Total demand is equal to 18, 000 and so each firm sells 9 , 000 units. T Ri = 9 , 000 × 5 = 45, 000 T Ci = 9 , 000 × 2 = 18, 000 Πi = 45, 000 − 18 , 000 = 27, 000 ∀i = α, β (c) {High price, Low price}. Firm α sells 2, 000 and while firm β sells 15, 000 units. T Rα = 2 , 000 × 10 = 20, 000 T Cα = 2 , 000 × 2 = 4, 000 Πα = 20, 000 − 4 , 000 = 16, 000 T Rβ = 15 , 000 × 5 = 75, 000 T Cβ = 15 , 000 × 2 = 30, 000 Πβ = 75, 000 − 30 , 000 = 45, 000
Game Theory
Solutions & Answers to Exercise Set 2
Giuseppe De Feo
May 10, 2011
Exercise 1 (Cournot duopoly) Market demand is given by
P (Q) =
140 − Q if Q < 140 0 otherwise
There are two firms, each with unit costs = £ 20. Firms can choose any quantity.
- Define the reaction functions of the firms;
- Find the Cournot equilibrium;
- Compare the Cournot equilibrium to the perfectly competitive outcome and to the monopoly outcome.
- One possible strategy for each firm is to produce half of the monopolist quantity. Would the resulting outcome be better for both firms (Pareto dominant)? Explain why this is not the equilibrium outcome of the Cournot game.
Solution.
- In a Cournot duopoly the reaction function of Firm A identifies its optimal response to any quantity produced by Firm B. In the presence of private firms, the optimal quantity is the one that maximizes ΠA , Firm A’s profit, where
ΠA = P (Q)qA − cqA = (140 − qA − qB ) qA − 20 qA
The first order condition for profit maximization is: ∂ΠA ∂qA^ =^140 −^2 qA^ −^ qB^ −^ 20 = 0 qˆA (qB ) = 120 − qB 2
Since the game is symmetric (firms have identical cost functions), the reaction function of firm B is: qˆB (qA ) = 120 − qA 2
- Cournot equilibrium is identified by the quantities that are mutually best responses for both firms; so, they are obtained by the solution of the following two-equation system: { qA = 120 − 2 qB qB = 120 − 2 qA The equilibrium quantities are q A? = q? B = 40 Q?^ = 80 and the equilibrium price is
P (Q?) = 140 − 80 = 60
and firms’ profits are: Π? B = Π? A = 60q? A − 20 q A? = 1600
- The competitive equilibrium outcome is characterized by P (Q) = c = 20. So total quantity should be:
P (Q) = 140 − Q = 20 Q = 120
In such a case firms’ profits are zero. The quantity produced by a monopoly is obtained by the usual first order condition:
ΠM = (140 − qM ) qM − 20 qM ∂ΠM ∂qM^ =^140 −^2 qM^ −^ 20 = 0 qM = 60
The price under monopoly is P (qM ) = 140 − 60 = 80 and profits are
ΠM = 80 × 60 − 20 × 60 = 3600
So, profits under monopoly are higher than the sum of firms’ profits under cournot competition; i.e., ΠM > Π? A + Π? B.
- If the each firm agreed to produce half of the monopolist quantity ( qM 2 = 40), their profits would be Π 2 M = 1800, larger than the Cournot profits. So, a Pareto improvement with respect to Cournot equilibrium would be possible. However, this cannot be an equilibrium since firms’ strategies are not mutually consistent best response; that is
qˆi
( (^) q M 2
6 = qM 2 ∀i = A, B
- Cournot equilibrium is identified by the quantities that are mutually best responses for both firms; so, they are obtained by the solution of the following two-equation system: { qA = 20 − 2 qB qB = 40 − 4 qA The equilibrium quantities are
q? A =^40 7 ; q? B =^60 7 Q?^ = 1007 and the equilibrium price is
P (Q?) = 40 −
and firms’ profits are:
Π? A = 180 7 q? A − 20 q A? =^1600 49 Π? B = 1807 q? B − q? B^2 =^720049
1 Bertrand oligopoly
Exercise 3 (Competition `a la Bertrand) Market demand is given by
P (Q) =
100 − Q if Q < 100 0 otherwise
Suppose that two firms both have average variable cost c = $50. Assuming that firms compete in prices, then:
- Define the reaction functions of the firms;
- Find the Bertrand equilibrium;
- Would your answer change if there were three firms? Why?
Solution.
- The construction of the reaction function in a competition `a la Bertrand proceeds in the following manner. Since firms compete in prices, we need to use the direct demand function where Q is a function of P. From the inverse demand:
Q (P ) =
100 − P if P < 100 0 otherwise
There are two firms, firm 1 and firm 2. Consider now the effect of the price choice of any firm i on its own profits for any given price chosen by firm j with i, j = 1, 2 and i 6 = j. (a) Pi > Pj : Πi = 0 In this case, firm 1 sells no output and therefore gets zero profits. (b) Pi = Pj : Πi = 12 [(Pj − 50) (100 − Pj )] This is where firm i shares the total profits in the industry. The total profits of the industry are here determined in the following way. Price minus average cost gives the profit per unit of output (Pj − 50) and this is multiplied by the total output (100 − Pj ). (c) Pi < Pj : Πi = [(Pi − 50) (100 − Pi)] here firm i gets the whole profits of the industry.
In order to firm the reaction function of firm i we can distinguish 3 different cases. (a) Pˆi (Pj ) = 50 if Pj ≤ 50 In such a case, any price Pi < Pj will make negative profits, so firm i will prefer to lose the race rather than beating j on price. More precisely we can say that firm i will never set a price below c = 50. This is usually portrayed as the strategy that if Pj ≤ 50, Pi = 50.^1 (b) Pˆi (Pj ) = Pj − if 50 < Pj ≤ 75 = P M where P M^ is the monopolistic price. If Pj > 50, there is potential for positive profits, as long as Pj < 100, at which point quantity demanded falls to zero, so that profits would be zero too. The first issue is, should firm i match the price of firm j, or attempt to undercut its rival? Intuition suggests that the firm should undercut its rival rather than match Pj. This can be shown formally by comparing the profits for firm i if it matches Pj , ΠAi , with the profits, ΠBi , it gets if undercuts Pj by a small amount so that Pi = Pj − ,.
ΠAi = 1 2 [(Pj − 50) (100 − Pj )] ΠBi = [(Pj − − 50) (100 − Pj + )] ΠBi = 2ΠAi − (150 − 2 Pj − 2 ) (^1) From a more formal game-theoretic point of view Pi = 50 is only one of the possible infinite best responses to Pj ≤ 50. However, for the sake of finding the Bertrand equilibrium, there is no loss in considering only the strategy Pj = 50 as best response to Pj ≤ 50.
where p¯ is the average price that is taken over the prices of the two firms. Each firm has average (and marginal) cost c = 20. Suppose the firms can only choose between the three prices { 94 , 84 , 74 }.
- Compute the profits of the firms under the 9 different price combinations that are possible in the model.
- Using you answer to the previous point, construct the 3x3 payoff matrix for the normal form game where the payoffs are given by the profits of the firms
- Find the (Bertrand-)Nash equilibrium of this game. What are the profits at this equilib- rium?
Solution. The profits of both firms will depend on both prices and can be written in the following way:
Π 1 (p 1 , p 2 ) =
180 − p 1 −
p 1 − p 1 + p 2 2
(p 1 − 20)
Π 2 (p 1 , p 2 ) =
180 − p 2 −
p 2 − p^1 + 2 p^2
(p 2 − 20)
Substituting for the different firms’ price choices profits are obtained and reported in the payoff matrix of the Bertrand game with discrete choices. It is easy to show that both firms
Firm 1
Firm 2 p 2 = 74 p 2 = 84 p 2 = 94 p 1 = 74 5724 , 5724 5994 , 5824 6264 , 5624 p 1 = 84 5824 , 5994 6144 , 6144 6464 , 5994 p 1 = 94 5624 , 6264 5994 , 6464 6364 , 6364
have a pi = 84 as a dominant strategy, and so the (Bertrand-)Nash equilibrium is given by the pair of strategies { 84 , 84 } and by the payoffs { 6144 , 6144 }.
Game Theory
Solutions & Answers to Exercise Set 3
Giuseppe De Feo
May 10, 2011
Exercise 1 (Sustainable cooperation in the long run) Two farmers, Joe and Giles, graze their animals on a common land. They can choose to use the common resource lightly or heavily and the resulting strategic interaction may be described as a simultaneous-move game. The payoff matrix is the following:
Joe
Giles light heavy light 40 , 40 20 , 55 heavy 55 , 20 30 , 30
- Find the Nash equilibrium of the game and show that it is an example of “Prisoners’ Dilemma” games.
- Suppose that the same game is repeated infinitely. Is the {light, light} outcome a Nash equilibrium if both players play a Grim strategy and have a discount factor of 0.7?
Solution.
- The Nash equilibrium is {heavy, heavy} with a payoff of 30 for both players. Indeed, both Joe and Giles have a dominant strategy to use the common land heavily. In addition, the Nash equilibrium is Pareto dominated by the outcome {40,40} arising when both players choose the dominated strategy light. The two features are characteristic of Prisoners’ Dilemma games. The game describes the so-called “tragedy of commons” in which the users of a common resource have an incentive to over-use it.
- When the game is repeated infinitely the trigger strategy is the following:
- Start playing cooperatively (light)
- Play cooperatively as long as the other player chooses light
Exercise 2 Distinguish simultaneous-move games and dynamic games in terms of information. Ex- plain why in dynamic games Nash equilibria may not be subgame perfect. Using examples, show how non-credible threats are ruled out using backward induction.
Solution. Students should discuss the difference between complete and perfect information in order to answer to the first question properly. Nash equilibria are not always subgame perfect because the concept on Nash equilibrium does not take into account the temporal dimension of the decision process in a dynamic game. So, menaces that at the beginning of the game might be used to threaten the opponent, would never be used when the time comes. Subgame perfection can therefore be considered a refinement applied to the Nash equilibrium concept that selects equilibrium strategies that are credible in every subgame of the whole game. The entry game is a good example to show the way in which backward induction works.
1 Finitely repeated games
Exercise 3 (Entry Deterrence) A market is characterized by a demand function Q = 1 − P and by a single firm with constant marginal cost c. The monopolist is facing potential entry from a new firm having the same marginal cost but an additional fixed cost of entry F = 0. 1. If the incumbent accepts the entry passively, then Cournot competition is played. However, the monopolist can threaten to produce the competitive output (i.e., the quantity such that P = c) so that the new entrant will make losses if it enters the market. If the new firm does not enter, the incumbent behaves as a monopolist.
- Assume that c = 0 and compute the payoffs of both firms (i.e., profits) in the cases of Monopoly, Cournot duopoly, and aggressive behaviour.
- Using the extensive form (game tree) representation, describe this entry game as a two- stage game where in the first stage the new entrant decides whether to enter or not, and in the second stage the incumbent firm decides whether to be passive or aggressive in case of entry, while it does nothing in case the new firm does not enter the market.
- Is the threat of aggressive behaviour by the monopolist credible? Answer the question by finding the subgame perfect equilibrium of the game.
- Describe the dynamic game using the normal form (pay-off matrix) representation and find the Nash equilibria of the game. Does the game have any Nash equilibria that are not subgame perfect?
Assume that incumbent is monopolist in 10 different national markets and it faces the threat of entry in all the markets sequentially (i.e., in stage 1 of the game the monopolist plays the game with a potential entrant in the Belgian market; in stage two the same game is played with another potential entrant for the Dutch market, and so on up to 10 rounds).
- Find the SPNE of the entire dynamic game by backward induction.
- Is there room for building a reputation? Explain your answer.
Solution.
- There are three possible outcomes of the game depending on the choices of the new entrant (entry vs no entry) and of the incumbent firn (aggressive vs passive behaviour). In case of no entry, the incumbent will set the monopolist level of output. Let i be the incumbent label and e the new firm label. The monopolist quantity is qim = 12 and the payoffs are Πmi = 14 and Πme = 0 (you should always show how to get there). If the new firm enters the market and the incumbent plays passively, Cournot competi- tion arises with qci = qce = 13 and Πci = 19 while Πce = 19 − 101 = 901. The two firms produce the same quantity but earn different profits given the fixed cost the new firm has to pay in order to enter the market. Finally, if the new firm enters the market and the incumbent behaves aggressively (i.e., it produces the quantity suc that p = c), then the new entrant pays the fixed cost but it is not able to produce a positive quantity since firm i serves all the market at p = 0. So, qai = 1, qae = 0, Πai = 0 and Πae = − 101.
Enter^ Stay out
E
Passive
{ 901 , 19 }
Aggressive
{− 101 , 0 }
I 1 Do nothing
{ 0 , 14 }
I 2
- The figure above depicts the extensive form of the two-stage dynamic game of entry deterrence we are analyzing.
- To assess the credibility of of the aggressive behaviour we rely on the notion of Subgame Perfect Nash Equilibrium. This is the Nash equilibrium of the dynamic game in which the equilibrium strategies of the players are optimal in each subgame since in each subgame the players are maximizing their payoff. In order to find the SPNE we use the backward induction technique by analyzing the last subgame, solving for the best choices, and going backward. Starting from the node I 1 (that is, we are assuming that in the first stage the new firm enter the market) the incumbent firms strictly refers to be passive rather than aggressive;
of stages) it might be possible (and useful) for the incumbent to build a reputation of tough player.
Exercise 4 (Centipede Game) Trinny and Susannah are playing the ”Centipede” game. At each node in the game the player can either move down, which means the game stops, or move right, which means that the game is passed to the other player. The two cycle game is depicted in Figure 1. Trinny’s pay-off is given first.
R R R
D D D
T 1 S 1 T 2
Figure 1:
- Solve the game by backward induction
- Show that the outcome is inefficient.
- If the game was extended to 100 cycles, with the pay-offs increasing in a similar manner, as represented in Figure 2, how would this affect the outcome of the game?
R R R
D D D
(100,98) (98,100)^ (101,99)
T 99 S 99 T 100
Figure 2:
Solution.
- We solve by going to the end and working backwards.
- At T2 Trinny plays down (D) as her pay-off of 3 will be greater than the correspond- ing pay-off of 2 when she plays right (R). We can therefore replace the sub-game at T2 with the pay-off (3,1).
- Next consider the Susannah’s decision at node S1. She will choose D, as her result- ing pay-off of 2 is greater then the pay-off of 1 if she passes the game to Trinny. Therefore we can replace the sub-game starting at S1 by the pay-off (0,2).
- Finally consider Trinny’s decision at T1. She gets a pay-off of 1 if she plays down and a pay-off of 0 if she passes the game onto Susannah. She therefore plays down.
The solution to the game is therefore that at T1 Trinny plays down (D) and the game ends with the pay-off (1,0)
- The outcome is clearly inefficient in that there are alternative outcomes that Pareto dominate the equilibrium outcome (that is, there are outcomes where both players receive a higher pay-off). An example would be if both players always played right, R, which gives a pay-off of (2,2).
- If the game was extended to 100 cycles, the outcome of the game should be exactly as previously: it finishes with the very first move with Trinny playing down, and a pay-off of (1,0). When it is played experimentally there is usually some co-operation though not right to the final round. How should Susannah interpret a move by Trinny at T1 to the right? This is an in- teresting question: how should one player interpret moves which are off the equilibrium path in a dynamic game? There are a number of possibilities. - One is that Trinny doesn’t understand the game. - A second is that Trinny understands the game but hasn’t fully worked out the implications (maybe there is some bounded rationality). - Third, it could be that Susannah is mistaken about Trinny’s pay-offs, and a move which appears irrational is in fact rational. For example, the pay-offs given at present might simply be the money pay-offs. But Trinny might get positive utility from Susannah getting a high money pay-off and therefore is prepared to pass the game to Susannah, even though that might mean that her own money pay-off is lower. This would be a form of altruistic behaviour. - Finally it might be that Trinny is offering co-operation even though this co-operative strategy appears to be unsustainable.
How Susannah should react will depend on which of these she thinks is correct.
