# Risk Dominance on a Network

Suppose there are players on the network illustrated on the right (with 1 player on each vertex). There are two players $H$ and $J$ that are on the ends of the green line. $H$ plays $A$ and player $J$ plays $B$. Players to the left of $H$ also play $A$ and to the right of $J$ each plays $B$.

Player $H$ deviating won't infect those to the left of $H$ iff $$x+w>y+z$$

Moreover, player $J$ defecting won't infect those to the right iff $$4y+z>4w+x$$

Does this mean that $A$ is risk dominant for those to the left of $H$ and $B$ for those to the right of $J$?

Risk dominance: if a player had a uniform prior over other players’ plays - so an equal chance of each other player playing A or B, then the risk dominant strategy would be the best expected payoff maximising choice for the player. I guess this is both satisfied for each case considered above. In the first case, one connection plays $A$ and the other $B$, so this gives a $0.5$ probability, and the risk dominant strategy should be played.