Game Theory with Applications to Marketing Strategy
Lecture 2: Nash Equilibrium
‧The strategy profile s* is a Nash equilibrium if no player has incentive to deviate from his strategy given that the other players do not deviate. Formally,
, , .
‧Find the pure strategy Nash equilibria for the games depicted in figure 1~3.
Figure 1 / Figure 2Figure 3
‧Hotelling model: Consumers are located uniformly along a linear city of length 1. Each consumer wants to buy one unit of a good from one of the existing firms. The transportation cost for the consumer is proportional to the distance to the firm he buys from. The law prohibits any form of competition through price or service (other than location), so consumers go to the nearest firm. A firm’s utility is equal to the number of its customers. Firms located at the same location get the same number of customers.
(i) There are two firms, and they choose their locations simultaneously. Show that there exists a unique pure strategy Nash equilibrium, and that both firms locate in the middle of the segment.
(ii) Show that there is no pure strategy NE with simultaneous moves for three firms.
(iii) Show that there is a pure strategy NE with simultaneous moves for four firms.
‧Consider pricingstrategy in Hotelling’s model
→ Reading: D'Aspremont, C., Gabszewicz, J. J., Thisse, J. F. (1979). On Hotelling’s “stability in competition”. Econometrica, 47(5), 1145-1150.
‧There are n consumers. Each consumer i spends an amount of money pi on a public good. The choices are simultaneous. Consumer i has utility
,
where g(0) = 0, g’(0) > 1, g’ > 0, g’’ < 0, and . Compute the Nash equilibrium. Discuss the multiplicity. Show that there is too little public expenditure.
‧Homogeneous good: Quantity competition (Cournot game) vs. Price competition (Bertrand game)
Cournot game: The set of players: two firms, i =1, 2. The strategies: two firms’ output quantities: , i =1, 2. The payoff of each player i:
,
where P(*) is the inverse demand function, c and F are respectively the variable and fixed costs. Suppose that c = F = 0 and P = 1 – q1 – q2. Find the Nash equilibrium.
Bertrand game: Two firms produce a homogeneous good and sell it to homogeneous consumers. Producing one unit of the good costs c, where 0 < c < 1. Each consumer is willing to pay as much as 1 dollar to buy 1 unit of the good. The total population of consumers is (normalized to) 1. The firms are competing in price. Consumers maximize consumer surplus and firms maximize profits. A pure strategy Nash equilibrium is (p1; p2), where pi is the price chosen by firm i, i = 1; 2. Assume that firms get the same market share if they choose the same price. Show that there is a unique Nash equilibrium (called the Bertrand outcome) where p1 = p2 = c.
The impacts of capacity constrain, e.g. .
How can firms with homogenous good alleviate the competition?
‧Differentiated good: Quantity competition (Cournot game) vs. Price competition (Bertrand game)
Two firms 1 and 2 with differentiated products facing the following demand curves:
, i =1, 2.
Compare the Nash equilibria under the assumption of price competition and quantity competition respectively. Do the firms compete more aggressively under price competition than quantity competition or vice versa? Interpret.
‧Consider the following two-player game. They each announce a natural number ai not exceeding 100 at the same time, i.e. 0≦a1, a2≦100 and a1, a2N. If a1+a2≦100, then π1 = a1 and π2 = a2. If a1+a2 > 100 and aiaj, then πi = ai and πj = 100 – ai. If a1+a2 > 100 and ai = aj, then π1 = π2 = 50. Determine if this game is dominance solvable.
‧Consider two firms A, B engaged in Cournot competition. The inverse demand is p = 3 – q1 – q2. Assume that both firms have marginal cost 1, but only firm A has the chance to spend F and brings the marginal cost down to zero. Firm A moves first by deciding to invest F or not to, which is unobservable to firm B (imperfect information). The two firms then play the Cournot game by selecting their own outputs. Compute the set of pure strategy Nash equilibria for this imperfect information game for . (Hint: for different ranges of F, the equilibria may differ)
‧There are 7 players. Each player is asked to pick a digit from the set {1, 2, …, 7}. Let xi be the digit selected by player i, and let be the simple average of xi’s. (Only) those players with xi closest to *0.8 will share 4200 dollars. Find all symmetric pure strategy Nash equilibria.
‧Payoff DominantRisk Dominant
A Nash equilibrium is considered payoff dominant if it is Pareto superior to all other Nash equilibria in the game.
A Nash equilibrium is considered risk dominant if it has the least loss.
Ranked Coordination: The equilibria can be Pareto ranked.
Dangerous Coordination
‧Focal points (Schelling, 1960)
Nash equilibria which for experiential or psychological reasons are particularly compelling.
Battles of the Sexes (趨同賽局)
Examples: 左轉右轉、先開槍再說？、古巴危機、兩岸對峙、訊息溝通的重要性、雨天收傘、VHS or Beta?、Blu-ray Disc or HD-DVD?
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