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I am reading a paper in the international journal of game theory entitled Unequal Connections by Goyal and Joshi (2006) and it has the following sentences: "If strategic complementarity obtains... In contrast, if substitutability holds..." I am not familiar with the terms "complementarity" and "substitutability" and a google search only turns up more articles which use the terms. What is the intuitive and/or mathematically rigorous definition for the two terms?

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You can look up the mathematical definitions in any game theory text book. The interpretation is basically that the players' actions reinforce/offset (complement/substitute) each other.

For example, if a buyer is more likely to buy a good at some price, when it is more likely that all other buyers are taking that price, then buyers are strategic complements. Alternatively, consider two firms. For some model parameters, it could be that one firm wants to produce less when the other firm produces more. Then, the firms are strategic substitutes.

Mathematically, you are interested in the sign of the cross derivative of the payoff function with respect to the players actions.

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