We have a basic game with signaling and subsequent moves. Before each player acts he receives some information about the game (part of the info is common to all players, some is not), by his acts he sends some information to other players.

The goal is to maximize $USD expectation (but it would be interesting to introduce CAPM-like model to account for variance and higher moments).

Before acting each player assigns some beliefs and (supposedly) acts on them (this is a tricky part, since it uses rationality assumption, which may fail in some situations), so we can construct Bayesian-Nash equilibrium for the game (we may ignore the fact that poker decision tree is infinitely large for simplicity reasons). With 3+ players I don’t believe it to be unique, but it’s extremely hard to prove.

It is important to formalize information structure, since it is basically the only thing that affects decisions of players.

Common prior:
1. Positions
2. Stack sizes
3. Payoff structure (for MTTs)
4. Board
5. Sizings, history of betting
6. Time to decide

Somewhere in between (non-common prior, but we may enter “k-level thinking game”, based on this information):
1. Stats, notes (easily accessible, help to form an idea about ranges, still easily manipulated and not precise)
2. Population tendencies
3. PIO suggestions (“modern disease of poker”, reference point, which affects the game of many)
4. Table dynamics (presence of weaker player is the most popular signal of this kind, but it covers broad area of signals)

Believe (non-common prior):
1. Ranges
2. Dead cards, board cards, randomization biases (superuser stuff:)