Best responses
Optimize against a stated opponent distribution and see where the preferred action changes.
11 minutes · advanced · Original Poker4Quant lessonWhat you will learn
Calculate action values as functions of an opponent mixture and identify ties.
Prerequisites: Dominance · Sensitivity and model error
Weight each action’s row by the same opponent distribution
If the column player chooses L with probability q and R otherwise, a row [a,b] has expected payoff qa+(1−q)b. Use the same q for every candidate action when comparing them. A best response maximizes that expectation.
The distribution is a hypothesis or a strategy description. Treating an estimated q as known can hide uncertainty about the best action. The payoff difference between two actions is often a simple linear function whose zero marks the switching threshold.
Randomization does not beat the best pure action against a fixed mixture
For a fixed opponent mixture, the expected payoff from mixing your actions is a weighted average of their action values. It cannot exceed the maximum of those values. If one action is strictly better, a best response uses it with probability one.
If several actions tie at the maximum, any mixture supported on those actions is also a best response. This does not mean arbitrary mixing creates a strategic equilibrium: the opponent’s incentives must be checked too.
Worked example
Rows A=[4,−2] and B=[1,1] face L with probability q. When is A a best response?
- EV(A)=4q−2(1−q)=6q−2.
- EV(B)=1 for every q.
- Equating values gives 6q−2=1, so q=1/2.
- A is better above one-half; B is better below; both tie at one-half.
At q=70%, A is worth 2.2 and B is worth 1. At q=20%, A is worth −0.8 and B is worth 1.
Common mistakes
- Using different opponent distributions when comparing rows.
- Assuming mixing improves an average beyond its best component.
- Calling a one-sided best response an equilibrium.
Keep these ideas
- Compute each action against one common opponent mixture.
- Find where payoff differences change sign.
- Tied best actions can be mixed without changing value.
Further reading
- MIT OpenCourseWare — Mixed Strategy Nash Equilibrium, lecture 3 ↗
University lecture background on expected utility, mixed strategies and indifference. Poker4Quant’s toy payoffs and exercises are original; this link does not imply affiliation or endorsement. Checked 2026-10-21.
- NYU — Game Theory, Problem Set 1 (2003) ↗
University problem set on best responses, dominance and the limitations of weak-dominance elimination. Background reading; the exercises on this site use their own specified payoffs. Checked 2026-10-21.