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.
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 $y>y'$.