What’s the Point?

The conclusion that players in a repeated prisoners’ dilemma game (PDG) will not cooperate at all depends on backward induction, which depends on the assumption of “common knowledge,” which is actually an assumption of infinite knowledge. First, A knows B is rational. Second, A knows B knows A is rational. And so on forever.

To understand why such an extreme assumption is needed, I will define a sequence of partial knowledges that, taken together, lead to common knowledge. Then we can examine the consequences of having various degrees of partial knowledge.

Notation and Definitions

“Rational” is defined as choosing to maximize the total expected payoff from a game or supergame.

• A»B|1 denotes “first-level” knowledge of player A about B.
• In particular, it denotes A’s knowledge that B is rational.
• Notated as:  A » BR,  so  A»B|1 == A » BR

Backtracking briefly, A»B|0 means A has no knowledge about B, which I will later interpret to mean A views B as selected at random from the general population which is 50% rational and 50% not. This is the case analyzed here. In this case (with “not rational” meaning a tit-for-tat player), it was found that in a particular two-stage PD supergame, the players cooperate in stage one, even though both are rational.

The next step will be to analyze a game in which players have first-level knowledge of each other.  Note that this means both are rational, and both know the other is rational. This “total knowledge set” is denoted by: 

• {K1} = {A»B|1 , B»A|1} == (A » BR)  &  (B » AR)

For later use, I will now define Kn. But first, here’s level 2.

• {A»B|2} == (A » BR)  &  (A » B » AR)
• {B»A|2} == (B » AR)  &  (B » A » BR)

These can be re-written in term of A’s and B’s knowledge sets:

• {A»B|2} == {A»B|1}  &  A » {B»A|1}
• Reverse A and B to find B’s level-2 knowledge set

Moving on to level 3, we have

• {A»B|3} == (A » BR)  &  (A » B » AR)  &  (A » B » A » BR)
• {B»A|3} == (B » AR)  &  (B » A » BR)  &  (B » A » B » AR)

The first of these says A’s level-3 knowledge of B is that: player A knows (1) that B is rational, (2) that B knows A is rational, and that (3) B knows, A knows B is rational.

These can be re-written in term of A’s and B’s knowledge sets:

• {A»B|3} == {A»B|1}  &  A » {B»A|2}
• Reverse A and B to find B’s level-3 knowledge set

And the general formula for level n is:

• {A»B|n} == {A»B|1}  &  A » {B»A|n-1}
• etc.

In general the n^th-level total-knowledge set is:

• Kn = { A»B|n , B»A|n },  for {player A , player B}

OR

• Kn = { BR & {B»A|n-1} , AR & {A»B|n-1} }

So, the n^th-level total-knowledge set includes: A knows B is rational and that B has an (n-1)^th-level knowledge of A, while B knows the reverse.

Where to from Here?

With 0-level knowledge sets we found that cooperation continued until the final stage of the PD supergame. I believe that with 1^st-level knowledge sets, it will continue until the final two stages, and so on through higher levels.

The 1^st-level result is interesting because it says that even though both players are rational and both know the other is, that is not enough to stop the coordinating except on the final two rounds of play. In a 100-stage supergame, they will cooperate for the first 98 rounds.

It is important to remember that if the total knowledge set is Kn, neither player has that knowledge. But have one-layer less knowledge of their opponent. If the game is analyzed as if they had Kn the discovered equilibrium will be wrong.