Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am given a 'tariff' function for two countries, $i=1, 2$. Both players can select a tariff between 0 and 100. If player $i$ selects $x_i$ and player $j$ selects $x_j$, country $i$ gets a payoff of $$2000+60x_i+x_ix_j-{x_i}^{2}-90x_j.$$ I need to find a Nash equilibrium for this game. Taking a derivative with respect to player 1, I get $$\pi_1'=60+x_2-2x_1,$$ so the payoff is maximized when $$x_1=\frac{60+x_2}{2}.$$ Symmetrically, for $x_2$, $$x_2=\frac{60+x_1}{2}.$$ Substituting one equation into the other, we get $x=60$ for both players. This makes sense, as both countries' lowering the tariff, to, say, 30 would increase their profits (which is something mentioned by the textbook -- equilibrium not necessarily optimal). My question is, what exactly is the meaning of the profit function for player 1 differentiated with respect to $x_2$? I get $x_1=90$ if I set the derivative equal to $0$, and I am wondering if this has any significance within the game theoretic model.

share|cite|improve this question
If one country pegs $x_1$ at 90, then the profit earned does not depend at all on what the other country does. – Willie Wong Apr 19 '11 at 12:38
up vote 3 down vote accepted

The partial derivative $\frac{\partial p_1}{\partial x_2}$ gives exactly what you might think: the change in player 1's payoff as player 2 changes their strategy. If this is zero, then player 1 will be indifferent to infinitesimal changes in player 2's strategy. If it is also true that player 1 is in a local payoff maximum (i.e. $\frac{\partial p_1}{\partial x_1}=0$ then, up to first order, a change in player 2's strategy will not be met by a corresponding change in player 1's strategy.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.