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One requirement for a supermodular game $(I, \mathbf S, \mathbf u)$ is usually presented in two ways (e.g. in this note):

For $i \in I$, $u_i$ is supermodular in $S_i$, when $s_{-i}$ is fixed, i.e. for $s_i, s_i' \in S_i$ $$u_i(s_i \vee s_i', s_{-i})+u_i(s_i \wedge s_i', s_{-i}) \geq u_i(s_i , s_{-i})+u_i(s_i', s_{-i})$$

Or

For $i \in I$, $u_i$ is upper-semicontinuous, i.e. for $\alpha \in \mathbb R$ $$\{s_i \in S_i \mid u_i(s_i, s_{-i}) < \alpha\} \text{ is open.}$$

But they're obviously different. The former endows $S_i$ with an order structure, while the latter requires the equipment of a topology. How to understand this difference?

Braiam
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Epicurus
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1 Answers1

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Your observation about topology and order is somewhat inaccurate. First definition make use of partial order on $S_i$ while the second make use of topology on it, but both structures are almost always specified by the assumption, that $S_i$ is a sublattise of $R^n$ and so it is equipped with usual subspace topology and partial order. This is the first point in the definition in the material you have referred to. So this worries are misplaced.

However there is some indications (see, for example, chapter 4 in Supermodularity and Complimentarity by Topkis), that the requirement of upper-semicontinuity is stronger and implies the first one. I'm not ready to provide an accurate proof, however.

Nikita Toropov
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  • Thank you for your reminder. I checked the index of the book you mentioned, but didn't find the key word upper-semicontinuity. – Epicurus Jan 10 '15 at 16:06
  • For example in that book in Lemma 4.2.2 and Theorem 4.2.1 impose an additional requiremen of upper semicontinuity on supermodular games which is, as I noted, a good indication of it being stronger requirement. And the fact, that it is sometimes defined that way suggest that upper semiconinuity is implied. – Nikita Toropov Jan 10 '15 at 18:21
  • Thank you for your reference and comment. It seems to me there's some confusion in defining supermodular games. Because as far as "The set of Nash equilibria of a supermodular game is a complete lattice" goes a la Zhou Lin, upper- semicontinuity is encompassed in the definition of a supermodular game. But in the source you cited, upper-semicontinuity is introduced as an additional requirement to obtain the nice result by Zhou. But it eludes me why upper-semicontinuity plus other conditions, compactness and increasing difference implies a supermodular utility function. – Metta World Peace Jan 11 '15 at 10:16