Description
Return (d-b)/(a-b-c+d) as opponentFirstProbability and whether it lies in [0, 1].
A two-action game can be represented by a small payoff matrix, but the interpretation depends on which player chooses each axis and whether their interests are opposed or coordinated. Indifference and a pure saddle are different conditions.
Return (d-b)/(a-b-c+d) as opponentFirstProbability and whether it lies in [0, 1].
When to use this tool
Use Mixed Strategy Two Action Indifference when you need a stated two-action payoff identity or classification.
How to use the tool and how it works
- Enter a, b, c, d. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read opponentFirstProbability, validProbability. Inspect validProbability before interpreting the associated values.
Require a-b-c+d nonzero. Return (d-b)/(a-b-c+d) as opponentFirstProbability and whether it lies in [0, 1].1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| a | A | Required. Finite number. |
| b | B | Required. Finite number. |
| c | C | Required. Finite number. |
| d | D | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| opponentFirstProbability | Indifference solution; may be outside [0, 1], so inspect validProbability. | Required. Finite number. |
| validProbability | Valid Probability | Required. Boolean. |
{
"input": {
"a": 4,
"b": 0,
"c": 1,
"d": 3
},
"output": {
"opponentFirstProbability": 0.5,
"validProbability": true
}
}Limitations and assumptions
- Rows are the focal player's actions in [[a,b],[c,d]]. An out-of-range solution is returned with false, and indifference alone does not establish a full equilibrium.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Solve the full game with its intended payoff conventions when equilibrium is the question. Check all best responses and feasibility conditions rather than inferring equilibrium from a single local equality.
References
Similar or alternative tools
- Zero Sum Two By Two Saddle Gap
For the supplied row-player payoff matrix [[a,b],[c,d]], return maximum row minimum, minimum column maximum, their gap, and exact equality as hasSaddle.
- Coordination Game Mixed Equilibrium Payoff Gap
Return B/(A+B), AB/(A+B), and that modeled mixed payoff minus outsidePayoff.