Strategic complements

Mathematically, consider a symmetric game with two players that each have payoff function ${\displaystyle \,\Pi (x_{i},x_{j})}$, where ${\displaystyle \,x_{i}}$ represents the player's own decision, and ${\displaystyle \,x_{j}}$ represents the decision of the other player. Assume ${\displaystyle \,\Pi }$ is increasing and concave in the player's own strategy ${\displaystyle \,x_{i}}$. Under these assumptions, the two decisions are strategic complements if an increase in each player's own decision ${\displaystyle \,x_{i}}$ raises the marginal payoff ${\displaystyle {\frac {\partial \Pi _{j}}{\partial x_{j}}}}$ of the other player. In other words, the decisions are strategic complements if the second derivative ${\displaystyle {\frac {\partial ^{2}\Pi _{j}}{\partial x_{j}\partial x_{i}}}}$ is positive for ${\displaystyle i\neq j}$. Equivalently, this means that the function ${\displaystyle \,\Pi }$ is supermodular.

On the other hand, the decisions are strategic substitutes if ${\displaystyle {\frac {\partial ^{2}\Pi _{j}}{\partial x_{j}\partial x_{i}}}}$ is negative, that is, if ${\displaystyle \,\Pi }$ is submodular.

In their original paper, Bulow et al. use a simple model of competition between two firms to illustrate their ideas. The revenue for firm x with production rates ${\displaystyle (x_{1},x_{2})}$ is given by

${\displaystyle U_{x}(x_{1},x_{2};y_{2})=p_{1}x_{1}+(1-x_{2}-y_{2})x_{2}-(x_{1}+x_{2})^{2}/2-F}$

while the revenue for firm y with production rate ${\displaystyle y_{2}}$ in market 2 is given by

${\displaystyle U_{y}(y_{2};x_{1},x_{2})=(1-x_{2}-y_{2})y_{2}-y_{2}^{2}/2-F}$

At any interior equilibrium, ${\displaystyle (x_{1}^{*},x_{2}^{*},y_{2}^{*})}$, we must have

${\displaystyle {\dfrac {\partial U_{x}}{\partial x_{1}}}=0,{\dfrac {\partial U_{x}}{\partial x_{2}}}=0,{\dfrac {\partial U_{y}}{\partial y_{2}}}=0.}$

Using vector calculus, geometric algebra, or differential geometry, Bulow et al. showed that the sensitivity of the Cournot equilibrium to changes in ${\displaystyle p_{1}}$ can be calculated in terms of second partial derivatives of the payoff functions:

${\displaystyle {\begin{bmatrix}{\dfrac {dx_{1}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dx_{2}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dy_{2}^{*}}{dp_{1}}}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial ^{2}U_{x}}{\partial x_{1}\partial x_{1}}}&{\dfrac {\partial ^{2}U_{x}}{\partial x_{1}\partial x_{2}}}&{\dfrac {\partial ^{2}U_{x}}{\partial x_{1}\partial y_{2}}}\\[2.2ex]{\dfrac {\partial ^{2}U_{x}}{\partial x_{1}\partial x_{2}}}&{\dfrac {\partial ^{2}U_{x}}{\partial x_{2}\partial x_{2}}}&{\dfrac {\partial ^{2}U_{x}}{\partial y_{2}\partial x_{2}}}\\[2.2ex]{\dfrac {\partial ^{2}U_{y}}{\partial x_{1}\partial y_{2}}}&{\dfrac {\partial ^{2}U_{y}}{\partial x_{2}\partial y_{2}}}&{\dfrac {\partial ^{2}U_{y}}{\partial y_{2}\partial y_{2}}}\end{bmatrix}}^{-1}{\begin{bmatrix}-{\dfrac {\partial ^{2}U_{x}}{\partial p_{1}\partial x_{1}}}\\[2.2ex]-{\dfrac {\partial ^{2}U_{x}}{\partial p_{1}\partial x_{2}}}\\[2.2ex]-{\dfrac {\partial ^{2}U_{y}}{\partial p_{1}\partial y_{2}}}\end{bmatrix}}}$

When ${\displaystyle 1/4\leq p_{1}\leq 2/3}$,

${\displaystyle {\begin{bmatrix}{\dfrac {dx_{1}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dx_{2}^{*}}{dp_{1}}}\\[2.2ex]{\dfrac {dy_{2}^{*}}{dp_{1}}}\end{bmatrix}}={\begin{bmatrix}-1&-1&0\\-1&-3&-1\\0&-1&-3\end{bmatrix}}^{-1}{\begin{bmatrix}-1\\0\\0\end{bmatrix}}={\frac {1}{5}}{\begin{bmatrix}8\\-3\\1\end{bmatrix}}}$

This, as price is increased in market 1, Firm x sells more in market 1 and less in market 2, while firm y sells more in market 2. If the Cournot equilibrium of this model is calculated explicitly, we find

${\displaystyle x_{1}^{*}=\max \left\{0,{\frac {8p_{1}-2}{5}}\right\},x_{2}^{*}=\max \left\{0,{\frac {2-3p_{1}}{5}}\right\},y_{2}^{*}={\frac {p_{1}+1}{5}}.}$

A game with strategic complements is also called a supermodular game. This was first formalized by Topkis,[3] and studied by Vives.[4] There are efficient algorithms for finding pure-strategy Nash equilibria in such games.[5][6]

• Supermodular
• Coordination game
• Coordination failure (economics)
• Uniqueness or multiplicity of equilibrium
• Multiplier (economics)