# Meaning of a partial derivative here?

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.

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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

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.