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AliceBob
Pick a cell. Nudge a number. Watch the game flip.
The whole point of game theory is this: outcomes don't come from what you want — they come from what both of you want, aware of each other. One well-chosen number flips the entire story.
A payoff matrix is embarrassingly small — four cells for a 2×2 game — and yet it covers price wars, driving conventions, standards fights, brinkmanship and penalty kicks. Below is the live version. Click a preset, then edit any number and watch the Nash equilibria rearrange themselves in real time.
Part 1 · Pick a game, edit a number
Two players: 🧡 Alice (amber — she picks a row) and 🩵 Bob (cyan — he picks a column). The matrix below shows what each cell pays each of them. Amber bars mark cells Alice prefers; cyan bars mark cells Bob prefers. Where amber and cyan meet you get a green Nash equilibrium — a cell neither would deviate from unilaterally. The analysis panel below updates every time you touch a number.
The payoff matrix · everything editable, everything live
🩵 Bob picks a column ▼
🧡 Alice picks a row ▲
| Cooperate | Defect | |
| Cooperate |
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AliceBob
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| Defect |
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AliceBob
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AliceBob
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Live analysis
Try this. Pick Prisoner's Dilemma. Both players
"defect" is the Nash — but it pays only 1 each when "both cooperate" pays 3
each. Now click the 1 in the bottom-right (Alice's payoff
for mutual defection) and change it to 5. The dilemma
dissolves. The game became Stag Hunt — two equilibria, (coop, coop) is
now one of them. One number did all the work.
Part 2 · Now play 20 rounds against five minds
The Prisoner's Dilemma matrix never changes — betraying is always the one-shot best move. But what if you play 20 rounds against the same opponent, and they remember? Pick an opponent, hit Cooperate or Defect, and watch the two score cards diverge. The right strategy depends entirely on the mind sitting across from you.
Repeated Prisoner's Dilemma · 20 rounds, 5 opponents
🥇 You
0
the human
vs
🥈 Opponent
0
Tit-for-Tat
You
Opponent
Round 1 of 20 — your move
🤝 both cooperate → 3 / 3
🗡️ you defect, they cooperate → 5 / 0
🤝 you cooperate, they defect → 0 / 5
🗡️ both defect → 1 / 1
Try all five. Against Always Defect, defecting
every round is clearly optimal (you'd lose 100 - 20 if you cooperated).
Against Tit-for-Tat, defecting feels right once — but then every
future round is mutual betrayal. The highest total you can reach is
60, and only by cooperating every round against someone
who cooperates back. Betrayal is dominant in one shot, but
disastrous when repeated.
What game theory is really for
Not prediction. The theory asks three questions, each worth more than any specific equilibrium: who moves when (changing this is usually the biggest available lever), what makes a threat credible (commitment is the deliberate destruction of your own options), and can you change the game instead of playing it better. That last one is mechanism design — and it's usually where the real returns are.
Three honest limits worth keeping in your pocket: payoffs are usually guessed, equilibrium ≠ good (the Prisoner's Dilemma proves that), and equilibrium ≠ what people do (centipede-game players don't backward-induct). The theory excels at reasoning — it shows you which threats are hollow, where commitment pays, and when randomising is mandatory.
Related models
- 🏔 NK rugged landscape — strategy on a landscape of interacting decisions, cousin to this model's strategic interactions between players.
- 🕸️ Networks — strategic games played on networks produce entirely different equilibria than well-mixed games.
- 🌐 Markov chains — the state of a repeated game evolves like a Markov chain over strategies.
