Actions and payoffs

Specify who chooses, what they know and whose payoff a table contains.

11 minutes · advanced · Original Poker4Quant lesson

What you will learn

Read a two-player payoff matrix and separate a toy strategic model from a poker solution.

Prerequisites: Expected value as a weighted average · Conditional directions

A game needs more than a list of good moves

Specify players, available actions, timing, information and payoffs. In a simultaneous two-player matrix game, each player chooses without observing the other’s current action. A row player chooses a row and a column player chooses a column; the chosen cell determines the outcome.

A table of row-player payoffs does not identify the column player’s incentives unless their payoff is also stated. In a zero-sum toy game, the column payoff is the negative of the row payoff. Do not infer zero-sum structure from the appearance of chips alone.

Payoff conventions must be consistent

Payoffs may represent net chips from a common starting point or abstract utility units. Both can support a mathematical model, but changing the accounting origin between cells can change its meaning. Previously committed chips and external prizes need explicit treatment.

A tiny matrix deliberately removes most of poker: private cards, multiple streets, legal bet sizes and changing information. It can teach strategic reasoning without being a full-game strategy recommendation. Its conclusions apply to the specified model.

Worked example

A zero-sum toy table gives the row player A/L=3, A/R=−1, B/L=0 and B/R=2. What happens at A/L?

The row action is A and the column action is L.
The table’s row payoff at that intersection is +3.
Because zero-sum payoffs were explicitly specified, the column payoff is −3.

The payoff pair is (+3,−3). Without the zero-sum assumption, the column payoff would need separate information.

Common mistakes

Reading a table entry as both players’ payoff.
Ignoring whether an action was observed before the response.
Calling a small toy game a solution of hold’em.

Keep these ideas

State actions, information and timing.
Name whose payoff is displayed.
Keep conclusions conditional on the model.