Calculator Form
Example Data Table
| Player A / Player B | High Scan | Low Scan |
|---|---|---|
| High Energy | A: 0, B: 0 | A: 6, B: 2 |
| Low Energy | A: 2, B: 6 | A: 4, B: 4 |
This sample creates a balanced mixed result. Both players receive equal expected value across supported strategies.
Formula Used
Let Player A choose Row 1 with probability p. Let Player B choose Column 1 with probability q.
Player A is indifferent when:
q a11 + (1 - q) a12 = q a21 + (1 - q) a22
So the calculator solves:
q = (a22 - a12) / (a11 - a12 - a21 + a22)
Player B is indifferent when:
p b11 + (1 - p) b21 = p b12 + (1 - p) b22
So the calculator solves:
p = (b22 - b21) / (b11 - b21 - b12 + b22)
Expected payoff is calculated by multiplying each cell payoff by its joint probability. The joint probabilities are pq, p(1 - q), (1 - p)q, and (1 - p)(1 - q).
How To Use This Calculator
- Enter two strategy names for Player A.
- Enter two strategy names for Player B.
- Add Player A and Player B payoff values for every cell.
- Choose decimal precision for the final output.
- Enter custom probabilities if you want a trial strategy check.
- Press Calculate to show results below the header.
- Use CSV or PDF buttons to save the current report.
Article
Why Mixed Strategy Balance Matters
Mixed strategy balance appears when no player should stay predictable. Each player spreads choices across actions. This idea helps explain contests, resource races, and paired physical models. A mixed plan is useful when one clear action can be exploited. It also supports classroom demonstrations of competing forces, switching systems, and strategic motion.
What This Calculator Does
This calculator solves a two by two payoff matrix. It accepts payoff values for both players. It also lets you name each strategy. The tool finds pure best responses. Then it checks whether an interior mixed equilibrium exists. When it exists, it reports both mixing probabilities. It also reports expected payoff values. Optional custom probabilities test any trial strategy pair.
Physics Style Use
Many physics examples contain rival choices. A robot may push or yield. A sensor may scan or save energy. Two agents may choose high or low effort. The numbers can represent energy, reward, loss, time, or stability. A mixed equilibrium can describe a stable randomized pattern. It does not prove nature is strategic. It gives a compact model for competing actions.
Reading The Result
Player A mixes between row strategies. Player B mixes between column strategies. The reported probability for Player A makes Player B indifferent. The reported probability for Player B makes Player A indifferent. If both values lie between zero and one, the mixed support is feasible. If not, the game may rely on pure equilibria or boundary behavior.
Practical Notes
Payoffs should use consistent units or scores. Larger values should mean better outcomes. Negative values are allowed. They often represent cost, force loss, or penalty. Avoid mixing utility scales without explanation. A tiny payoff change can move the equilibrium. So check sensitivity with nearby values.
Best Use
Start with a simple example. Confirm that strategy labels are clear. Enter all four payoff pairs. Review pure equilibrium cells first. Then inspect the mixed solution. Export results when you need records. Use the formula section to explain your work. Use the custom check to test proposed laboratory or game scenarios.
Limitations
The calculator assumes two players, two actions, and known payoffs. Larger games need support enumeration or numerical solvers. Use this result only as guidance.
FAQs
What is a mixed strategy Nash equilibrium?
It is a strategy pair where players randomize among actions. Each player’s probability choice makes the other player indifferent between supported actions. No player can improve expected payoff by changing alone.
Can this calculator solve every game?
No. It is built for two player, two strategy games. Larger games need other methods, such as support enumeration, linear programming, or numerical solvers.
Why is this listed for physics use?
Physics style models can include competing agents, energy choices, scanning actions, or control decisions. The calculator treats those choices as a payoff matrix for simplified analysis.
What do p and q mean?
p is Player A’s probability of choosing the first row strategy. q is Player B’s probability of choosing the first column strategy.
What happens when probability is outside zero and one?
The interior mixed equilibrium is not feasible. The game may have pure equilibria, boundary solutions, dominance, or no equilibrium on that selected support.
Can payoff values be negative?
Yes. Negative values are allowed. They can represent costs, losses, penalties, wasted energy, or unstable outcomes. Keep the meaning consistent across all cells.
What is the custom strategy test?
It checks any entered probability pair. It reports expected payoffs, best responses, and improvement gaps. This helps compare a trial strategy with the equilibrium result.
Why do CSV and PDF exports use the same inputs?
The export buttons calculate the current form values before download. This saves the same matrix, probabilities, payoffs, dominance checks, and response checks shown by the tool.