Shaking the Tree

Life often results in situations such that no strategy suggests any further moves. We just don’t know what to do next. In a game of perfect information, where each player knows all the previous moves, this can signal stalemate. Take chess: given both sides know everything that has transpired and have no reason to believe that the opponent will make a mistake, there can come a time when both sides will realize that there are no winning strategies for either player. A draw is then agreed upon.

The situation is not as simple in games of incomplete information. Let’s assume some information is private, that some moves in the game are only known to a limited number of players. For instance, imagine you take over a game of chess in the middle of a match. The previous moves would be known to your opponent and the absent player, but not to you. Hence you do not know the strategies used to arrive at that point in the game, and **your opponent knows that you do not know**.

Assume we are in a some such situation where we do not know all the previous moves and have no further strategic moves to make. This is to say we are waiting, idling, or otherwise biding our time until something of significance happens. Formally we are at an equilibrium.

A strategy to get out of this equilibrium is to “shake the tree” to see what “falls out”. This involves making information public that was thought to be private. For instance, say you knew a damaging secret to someone in power and that person thought they had successfully hidden said secret. By making that person believe that the secret was public knowledge, this could cause them to act in a way they would not otherwise, breaking the equilibrium.

How, though, to represent this formally? The move made in shaking the tree is to make information public that was believed to be private. To represent this in logic we need a mechanism that represents public and private information. I will use the forward slash notation of Independence Friendly Logic, /, to mean ‘depends upon’ and the back slash, , to mean ‘independent of.’

To represent private strategy Q, based on secret S, and not public to party Z we can say:

Secret Strategy) If, and only if, no one other than Y depends upon the Secret, then use Strategy Q
(∀YS) (∃z/S) ~(Y = z) ⇔ Q

To initial ‘shaking the tree’ would be to introduce a new dependency:

Tree Shaking) there is someone other than Y that depends on S
(∃zS) ~(Y = z)

Tree Shaking causes party Y’s to change away from Strategy Q since Strategy Q was predicated upon no one other than Y knowing the secret, S. The change in strategy means that the players are no longer idling in equilibrium, which is the goal of shaking the tree.

Posted in game theory, independence friendly logic, logic, philosophy. Tagged with , , .