Description
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.
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.
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.
When to use this tool
Use Zero Sum Two By Two Saddle Gap 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 maximin, minimax, gap, hasSaddle. Inspect hasSaddle before interpreting the associated values.
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.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| a | Row 1 / column 1 payoff to the maximizing row player. | Required. Finite number. |
| b | Row 1 / column 2 payoff to that player. | Required. Finite number. |
| c | Row 2 / column 1 payoff to that player. | Required. Finite number. |
| d | Row 2 / column 2 payoff to that player. | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| maximin | Maximin | Required. Finite number. |
| minimax | Minimax | Required. Finite number. |
| gap | Difference on the stated unit and subtraction basis. | Required. Finite number. |
| hasSaddle | Has Saddle | Required. Boolean. |
{
"input": {
"a": 1,
"b": -1,
"c": -1,
"d": 1
},
"output": {
"maximin": -1,
"minimax": 1,
"gap": 2,
"hasSaddle": false
}
}Limitations and assumptions
- The row player maximizes and column player minimizes the same payoff. This tests a pure saddle, not a mixed-equilibrium solution or a general-sum game.
- 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
- Mixed Strategy Two Action Indifference
Return (d-b)/(a-b-c+d) as opponentFirstProbability and whether it lies in [0, 1].
- Game Theory: Prisoner Dilemma Payoff Ordering Check
Test strict temptation>reward>punishment>sucker and separately 2×reward>temptation+sucker.