game theory

Handout 10

ECO 444

Konrad Grabiszewski

Bertrand Oligopoly

In the market, there are 2 firms producing a single good. Let i denote a generic firm. Each firm

decides how much to charge (price competition). Let pi denote the price of firm i.

Firms incur costs, and we assume a linear cost function for each firm; Ci(qi) = ciqi, where ci is

the marginal cost. We assume that firms have the same marginal costs; i.e., cA = cB = c.

If the firms set different prices, then the lowest price wins the whole market (i.e., firms with price

higher than the lowest price sell nothing). If the firms set the same price, then the demand is split

among them equally.

We assume a linear demand function. Let p = min{pA, pB}.

Q(p) =

  a−p if p ≤ a

0 if p > a

(1)

Normal-form representation.

• N = {A, B} is the set of players;

• Si = [0,∞) is the set of strategies of player i; a strategy pi is the price;

• ui is the utility function of player i defined as

ui(pA, pB) =

  piQ(pi) − cQ(pi) if pi < pj 1 2 piQ(pi) − 12cQ(pi) if pi = pj

0 if pi > pj

(2)

Note that piQ(pi) − cQ(pi) = (pi − c)Q(pi) = (pi − c)(a−pi).

First, we solve the problem of monopolist who chooses p to maximize (p − c)(a − p). Note that

the monopolist’s optimal price is p∗ = 1 2 (a + c).

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Equilibrium strategies. There is only one Nash equilibrium: (c, c). First, note that (c, c) is,

indeed, a Nash equilibrium. This is because no firm has a reason to unilaterally deviate from

(c, c). Second, we need to argue that (c, c) is the only Nash equilibrium:

1. None of the firms will price below c because this implies negative profits.

2. A profile (pi, pj) such that pi = c and pj > c is not a Nash equilibrium because firm i can

gain more by increasing their price a little bit.

3. A profile (pi, pj) such that pi ≥ pj > c is not a Nash equilibrium because firm i can gain

more by decreasing their price a little bit below pj.

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