# Submodular and supermodular games

Can someone please explain to me (with concrete examples) what are submodular and supermodular games, and their related concepts of games of strategic substitutes and strategic complements.

An artificial example using Prisoner's Dilemma would be quite helpful. Thanks in advance.

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The prisoners dilemma is not a very insightful example. The most accessible material I know on these topic are these notes, but it is still not easy. – Michael Greinecker Jul 17 '12 at 17:27
@MichaelGreinecker, I had a look at similar materials, and struggled to develop an intuitive understanding of the concepts, hence I am asking for a concrete example using a simple game to illustrate the concepts. But thanks for the notes anyway! – MLister Jul 17 '12 at 17:41

You have two players, Ann and Bob. Both have as their strategy spaces the unit interval $[0,1]$. Also $u_A(x,y)=u_b(x,y)=u(x,y)=xy$, the game is of common interest.

There are two pure-strategy equilibria, $(0,0)$ and $(1,1)$. The game has strategic complements, the payoff functions satisfy increasing differences. If $x>x'$ and $y>y'$, then $$u(x,y)-u(x',y)> u(x,y')-u(x',y)$$ since $(x-x')y> (x-x')y'$ when $(x-x')>0$ and $y>y'$.

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For simplicity, imagine that a strategy is just to pick up a number. Now think about the value of increasing the strategy of player 1 assuming player 2 strategy is being held constant. If the value of increasing the strategy is higher the higher is the strategy of player 2, we say the payoff of player 1 is super-modular in the strategies. If this happens for both players, their strategies are said to be complementary. If payoffs are differentiable this is the same as the cross-partial derivatives of payoffs being positive: $\frac{\partial u_i}{\partial s_1 s_2}(s_1,s_2)\ge 0$ for $i=1,2$.

In the case of ta finite game like the Prisoners Dilemma, we must somehow order the strategies. Let's say that cooperation is low strategy and defect is the high strategy, $C<D$. Super-modularity would imply that $u_1(D,C)-u_1(C,C) \le u_1(D,D)-u_1(C,D)$. That is, the gain of changing from cooperating to defecting given the opponent cooperates must be less than the gain of changing from cooperating to defect given the opponent defects. Some remarks: If you choose a different order (ay $C>D$) then you still have the same inequality as before. Not all Prisoners Dilemma games are super-modular.

$$D>C,\;\text{super-modular}\Longrightarrow \begin{matrix} & D & C \\ D & 0,0 & 3,-7 \\ C & -7,3 & 2,2 \end{matrix}$$

$$D>C,\;\text{not super-modular}\Longrightarrow \begin{matrix} & D & C \\ D & 0,0 & 7,-1 \\ C & -1,7 & 2,2 \end{matrix}$$

Super-modularity is more interesting/useful in the context of coordination or matching games than in the prisoners dilemma because it implies the set of Nash equilibrium is a lattice (nice math object). In the case of the prisoner dilemma, there is only one Nash equilibrium so we are not interesting in talking about the set of equilibria...

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