E(uA)?rs(1)?r(1?s)(?1)?(1?r)s(?1)?(1?r)(1?s)(1) ?4rs?2r?2s?1?2r(2s?1)?2s?1
Oviously, A’s optimal choice of r depends on B’s probability, s. If s?12, utility is maximized by choosing r?0. If s?12, A should opt for r?1. And when s?12, A’s expected utility is 0 no matter what value of r is choosen. A’s best response function（反应函数） is
?0, if s?12? r(s)??1, if s?12
?[0,1], if s?12?For player B, expected utility is given by
E(uB)?rs(?1)?r(1?s)(1)?(1?r)s(1)?(1?r)(1?s)(?1) ??(4rs?2r?2s?1)?2s(1?2r)?(1?2r)
Now, when r?12, B’s expected utility is maximized by choosing s?0. If r?12, A should opt for s?1. And when r?12, A’s expected utility is independent of what s is choosen. B’s best response function （反应函数）is
?1, if r?12? s(r)??0, if r?12
?[0,1], if r?12?sresponse curve of B1response curve of A1/2
1/21r
Nash equilibria are shown in the figure by the intersections of optimal response curves for A and B.
Or, we can get the equilibrium through the FOC
?E(uA)1?4s?2?0 ? s? ?r2?E(uB)1?4r?2?0 ? r?
?s2
3．第二种方法
在给定参与人B采用混合战略(s)H ? (1-s)T的情况下，如果混合战略(r)H ? (1-r)T是参与人A的最优选择，必有EuA(H)?EuA(T)。同样的，在给定参与人A采用混合战略
(r)H ? (1-r)T的情况下，如果混合战略(s)H ? (1-s)T是参与人B的最优选择，必有EuB(H)?EuB(T)。这样，混合策略Nash均衡就可以由以下两式得到
?即
??EuA(H)?EuA(T)
?EuB(H)?EuB(T)?1?s?(?1)?(1?s)?(?1)?s?1?(1?s)
(?1)?r?1?(1?r)?1?r?(?1)?(1?r)?这样很容易就可以得到上面的混合策略Nash均衡。 作业：求“性别之战”博弈的混合策略Nash均衡。
七、Nash均衡在寡头市场中的应用
1．古诺竞争
Suppose there are two firms in an industry. Their strategy spaces are quantities. Their payoffs are profits.
Industry demand is given by the inverse demand function, P(Q), where industry production is
Q?q1?q2. They have identical cost functions c(qi). Profits for each firm are therefore given
by:
?1?P(q1?q2)q1?c(q1) and ?2?P(q1?q2)q2?c(q2)
This is a game. The Nash equilibrium occurs when both firms are optimising given the behaviour of the other.
Throughout the lecture consider the following linear demand example with constant marginal costs. So:
P(Q)?a?Q?a?q1?q2 and c(q)?cq
Profits and Best Responses
Profits for each firm are maximised where marginal revenue is equal to marginal cost. Recall firms are interested in finding an optimal level of their quantity for each level their opponent might choose. Suppose firm 2 chooses q2:
max?1?max(a?q1?q2)q1?cq1
q1q1q2 acts like a constant. For linear demand, marginal revenue falls at twice the rate of demand.
Firm 1 sets marginal revenue, a?2q1?q2, equal to marginal cost, c. Hence:
q1(q2)?a?c?q2a?c?q1 and q2(q1)? 22These two equations give the best response functions for the two firms. Often called reaction or
best reply functions. Plotting these yield the reaction or best reply curves. Reaction Curves
Drawing the reaction curves for both firms on the same graph yields the picture below.
The curve q1(q2) yields the optimal level of q1 for any given q2. The curve q2(q1) yields the optimal level of q2 for any given q1. These curves will cross. Why?
A point at which there is no profitable unilateral deviation is a Nash equilibrium. That is a point which is a best response to a best response (and so on) — where the two curves cross, written
**(q1,q2).
Nash Equilibrium
**What is the value of q1 and q2 ? They could be read off from the graph. Alternatively, solve
the two equations.
Substituting the value of q2 into the equation for q1 yields:
*a?c?q11a?c*q?(a?c?) ? q1?
223*1**Symmetrically solving for q2 gives q2?(a?c)3.
With the same linear demand and constant marginal cost assumptions in place but with n firms in the industry it is (mathematically) simple to show that each of the n firms will produce at:
qi*?a?c n?1Equilibrium Profits and Prices
How much do the firms charge and how much profit do they make? Price is simply read off from the demand curve.
***Equilibrium industry supply is Q?q1?q2. Hence equilibrium price is:
**P*?P(Q*)?a?q1?q2?a?2c 3***2Profit is given by revenue less cost: ?1?(P?c)q1?(a?c)9.
It is just as easy to calculate the price and profit in the case of n firms:
P*?anca?c2* and ?i?(?)
n?1n?1n?1Notice that, as the number of firms grows, price gets closer to marginal cost and profits get close
to zero. These are the conditions in a perfectly competitive market. An oligopoly with many firms is like perfect competition. 2．合谋
How does the case of Cournot duopoly differ from monopoly? If the two firms could collude they would act like a monopoly to maximise total profits. Recall a monopolist faces the entire demand curve and sets MR = MC.
With linear demand P?a?Q and constant marginal costs c, the optimality condition requires:
MR?MC ? a?2Qm?c ? Qm?ma?c 2A monopolist produces less and hence prices higher at P?(a?c)2. Profits are
?m?(a?c)24.
If the two firms could collude they would be able to split the profits in two, each firm getting
?m?(a?c)28 by producing q1m?(a?c)4. This is bigger than their Cournot equilibrium
profit.
But they cannot. If one of the firms produced (a?c)4 the other would not choose to produce the same. The best response function reveals that the firm has a better response where if (for example) firm 1 produced (a?c)4:
q2(q1)?a?c?q11a?c3?(a?c?)?(a?c) 2248This will yield higher profits. How can collusion be explained?
3．伯特兰竞争
Suppose instead firms choose prices — does this make a difference?
Firm 1 and firm 2 choose p1 and p2 simultaneously. The firm that charges the lower price serves the entire market. If both firms charge the same price, they both serve half the market. Profits are payoffs. Writing down profits, suppose that market demand is Q(P) where P is market price (the lower of the two prices, p1 and p2). Suppose again that marginal costs are constant,
MC?c.
?p1Q(p1)?cQ(p1) if p1?p2?1??1??{p1Q(p1)?cQ(p1)} if p1?p2
?2?? 0 if p1?p2To illustrate the best reply functions consider what firm 1's optimal response is if firm 2 sets a price p2. If p2 is greater than marginal cost then firm 1 will wish to undercut firm 2 by a small amount.
If p2is equal to marginal cost, firm 1 would be willing to set any price greater than or equal to
p2— all such prices will result in zero profits. Firm 1 will never set a price below p2 if p2 is
less than c since this results in losses. In this last case, firm 1 would set any price strictly greater than p2 and get zero profits.
Reaction Curves and Nash Equilibrium
Drawing this argument in an informal way gives the below \
The only place where the two curves cross — the Nash equilibrium — is at (p1,p2)?(c,c). The firms price at marginal cost — the efficient outcome. It makes no difference how many firms
are in the market.
The basic idea is that the firms will continue to undercut one another until they reach marginal cost. They will go no lower as this would involve making a loss.
This is very different to Cournot. However, the Cournot equilibrium can be recovered with capacity constraints.
问题：如果MC1?MC2，均衡是什么？
CH10 博弈论与寡头垄断
