In the first series of the British game show Golden Balls, a contestant named Nick faced his final opponent, Ibrahim, with £13,600 on the table. Each had to secretly pick Split or Steal. Nick announced he was 100% going to steal and told Ibrahim to pick Split, promising to share the money after the show. Then Nick chose Split himself. Both went home with £6,800.
Game theory games are simplified models of situations where your result depends on your choice and someone else’s. The six classics are the Prisoner’s Dilemma, Stag Hunt, Chicken, Battle of the Sexes, Matching Pennies, and the Ultimatum game. Each isolates one problem: temptation, trust, nerve, compromise, predictability, or bargaining.
Nick’s move only makes sense once you see which game he was in, and how he quietly changed it.
Split or Steal shows that small rule changes create a different game
The rules of Split or Steal are simple. If both choose Split, they share the jackpot. If one steals and the other splits, the stealer takes everything. If both steal, both leave with nothing.
That looks like a Prisoner’s Dilemma, with one important difference. In the classic dilemma, defecting against a defector still beats cooperating. Here, if your opponent steals, you get zero whatever you pick. Steal is only weakly better: it never does worse and sometimes does better.
Nick used that tie. Once he convinced Ibrahim he would steal, Split cost Ibrahim nothing compared with Steal, and it kept alive the chance of a share after the show. In effect, Nick turned a simultaneous dilemma into something closer to an ultimatum offer. Ibrahim accepted, and Nick’s surprise Split made the promise unnecessary.
Real contestants do not behave like the textbook either. Economists Martijn van den Assem, Dennie van Dolder, and Richard Thaler analyzed the show’s final round, where stakes averaged over $20,000, in “Split or Steal? Cooperative Behavior When the Stakes Are Large”. Individual players chose Split 53% of the time. Cooperation was surprisingly high for sums that would normally matter but looked small next to the show’s bigger jackpots, which the authors called a “big peanuts” effect.
Trust did not always pay. In a 2008 special, a contestant named Sarah stole £100,150 from Stephen, who had chosen Split.
So the name of a game is shorthand for a specific payoff pattern. Change one cell, the timing, or what players can say, and you may be in a different game.
How to read a payoff matrix
A game specifies who chooses, what they can choose, and how each combination of choices affects each player’s payoff. A fuller description also says what players know, when they act, and whether they meet again. Yale’s introductory lecture explains why an outcome (money, a grade) only becomes a payoff once you know how the player values it.
In every square table below, Alex chooses a row and Blair chooses a column. Read each pair as (Alex’s payoff, Blair’s payoff). Larger numbers mean a preferred outcome for that player.
Follow one row and one column to their intersection. The first payoff belongs to the row player; the second belongs to the column player.
A cell is a pure-strategy Nash equilibrium if neither player could improve by changing only their own choice. That is the unilateral-change rule in MIT’s explanation of Nash equilibrium. An equilibrium need not have the highest combined payoff, and it need not treat both players equally.
The six classic games, one table each
These are teaching versions with illustrative numbers. Textbooks may use different numbers or stories, but the ordering of each player’s payoffs is what defines the game.
1. Prisoner’s Dilemma: individually tempting, jointly worse
Two players independently choose Cooperate or Defect.
| Alex / Blair | Cooperate | Defect |
|---|---|---|
| Cooperate | (3, 3) | (0, 5) |
| Defect | (5, 0) | (1, 1) |
If Blair cooperates, Alex gets 5 by defecting instead of 3. If Blair defects, Alex gets 1 instead of 0. Blair faces the same comparisons, so Defect is strictly dominant for both. The only pure equilibrium is (Defect, Defect), yet both would prefer (3, 3). That gap is the dilemma.
A shared project fits this pattern only if each person prefers to skip the work whatever the partner does. If you would rather help when the other person helps, it is a different game. The dominant strategy guide shows how the 1971 US ban on TV cigarette ads got an industry out of exactly this trap.
2. Stag Hunt: a better joint outcome needs confidence
Each player chooses Stag, which only pays if both join, or Hare, a safe solo option.
| Alex / Blair | Stag | Hare |
|---|---|---|
| Stag | (4, 4) | (0, 3) |
| Hare | (3, 0) | (3, 3) |
If Blair hunts stag, Alex prefers Stag (4 beats 3). If Blair hunts hare, Alex prefers Hare (3 beats 0). There are two pure equilibria, (Stag, Stag) and (Hare, Hare). Both prefer the first, but going for Stag alone gives the worst payoff.
A team switching to a new shared tool can look like this: staying put is fine, switching together is better, switching alone is costly. The key contrast with the dilemma is that here cooperation is your best response to cooperation. Cambridge’s lecture slides compare the two models.
3. Chicken: both want the other to give way
Players choose Yield or Hold. Each prefers holding while the other yields; both holding is the worst outcome.
| Alex / Blair | Yield | Hold |
|---|---|---|
| Yield | (2, 2) | (1, 3) |
| Hold | (3, 1) | (0, 0) |
The pure equilibria are (Yield, Hold) and (Hold, Yield). Mutual yielding at (2, 2) beats mutual holding, but it fails the unilateral test: either player gains by holding alone.
Two people insisting on incompatible meeting times can fit Chicken, but only if each would rather concede than lose the meeting entirely. The model does not say escalation is wise.
4. Battle of the Sexes: agree to meet, disagree on where
The traditional name describes a coordination game with conflicting preferences; gender plays no role. Alex and Blair want to go out together, but Alex prefers the Concert and Blair prefers the Match.
| Alex / Blair | Concert | Match |
|---|---|---|
| Concert | (2, 1) | (0, 0) |
| Match | (0, 0) | (1, 2) |
Both (Concert, Concert) and (Match, Match) are equilibria, since leaving alone drops the switcher to 0. The conflict is over which one.
Two coordinated outcomes are stable, but the players prefer different ones.
Unlike Stag Hunt, the players disagree about which coordinated outcome is best. Yale’s coordination lecture explores this tension. An agreed rule, such as taking turns, works by turning one round into a repeated game.
5. Matching Pennies: a predictable choice can be exploited
Each player secretly picks Heads or Tails. Alex wins when they match; Blair wins when they differ.
| Alex / Blair | Heads | Tails |
|---|---|---|
| Heads | (1, −1) | (−1, 1) |
| Tails | (−1, 1) | (1, −1) |
This game is zero-sum, and no cell is a pure equilibrium: at a match Blair wants to switch, at a mismatch Alex does. The equilibrium is a mixed strategy, here a 50/50 coin flip for each player, which leaves the opponent nothing to exploit.
Soccer penalty kicks work the same way, and professional players mix sides almost exactly as the theory predicts. The Nash equilibrium guide walks through that data. Note that alternating Heads and Tails is a pattern, and patterns can be read.
6. Ultimatum: the response comes after the offer
One player proposes how to split a resource; the other accepts or rejects. Acceptance carries out the split, and rejection leaves both with nothing. These are the rules in Yale’s bargaining lecture.
Suppose Alex proposes splitting 10 tokens, keeping 6 and offering Blair 4.
| Blair’s response to this offer | Alex’s tokens | Blair’s tokens |
|---|---|---|
| Accept | 6 | 4 |
| Reject | 0 | 0 |
One offer-response branch: Alex keeps 6 and offers Blair 4. Other possible offers are omitted.
This is a sequential game: Blair sees the offer before responding. If Blair cares only about tokens, accepting 4 beats rejecting. If fairness or reputation matters, tokens alone may not capture Blair’s payoff. Nick’s Golden Balls pitch borrowed this structure: commit first, then make the other side an offer.
The six games side by side
| Game | Pure equilibria | Core problem | What tends to change it |
|---|---|---|---|
| Prisoner’s Dilemma | One, worse for both | Temptation to defect | Enforceable rules, repetition |
| Stag Hunt | Two | Trust in the other’s commitment | Signals, reassurance |
| Chicken | Two | Nobody wants to back down | Credible commitment |
| Battle of the Sexes | Two | Agreeing on which outcome | Conventions, turn-taking |
| Matching Pennies | None | Being predictable | Genuine randomness |
| Ultimatum | Many; theory favors a minimal offer | Fairness vs. payoff | Outside options, norms |
Five features that classify any game
A game’s name identifies an incentive pattern. Its type describes features, and one game can have several.
- Zero-sum or not: Do payoffs always add to zero? Matching Pennies does; the Prisoner’s Dilemma does not.
- Simultaneous or sequential: Are choices hidden from each other, or does one player move after seeing the other’s move?
- Symmetric or asymmetric: Do the roles swap cleanly? The Prisoner’s Dilemma here is symmetric; Battle of the Sexes is not.
- Cooperative or noncooperative: Can players sign enforceable agreements? Choosing an action called “Cooperate” does not make a game cooperative in this technical sense.
- Perfect or imperfect information: Does each player see all earlier moves before choosing?
Also note whether the game is played once or repeatedly. The game theory guide shows how repetition let a simple strategy win Robert Axelrod’s tournaments.
A one-player-switch check you can run on any table
Take (Cooperate, Cooperate) in the Prisoner’s Dilemma.
| Check | Hold fixed | Switch | Own payoff change |
|---|---|---|---|
| Alex’s choice | Blair cooperates | Alex defects | 3 → 5 |
| Blair’s choice | Alex cooperates | Blair defects | 3 → 5 |
Both have a profitable solo switch, so this cell fails. Now check (Defect, Defect): holding Blair at Defect, Alex drops from 1 to 0 by switching, and the same holds for Blair. This cell passes, despite paying both less.
For any square table:
- Pick a cell and note both payoffs.
- Keep the column fixed and compare the row player’s options using the first number.
- Return to the original cell, keep the row fixed, and compare the column player’s options using the second number.
- If either player has a strictly better option, the cell is not a pure equilibrium.
This finds pure equilibria only. It does not compute mixed ones or predict what people will actually do.
When naming the game leads you astray
The label hides the payoffs. Calling a group project a “prisoner’s dilemma” assumes everyone prefers to shirk no matter what. If people actually want to help when others help, it is a Stag Hunt, and reassurance will do more than enforcement.
Real preferences are messier than the table. Golden Balls players split far more often than a pure money model predicts, and reacted to how their opponent had treated them earlier in the show.
Talk can change the game. Nick’s announcement changed what Ibrahim expected, and therefore what Ibrahim’s best response was.
Repetition changes the incentives. A one-shot table says little about colleagues who will deal with each other for years.
For tracing what happens after the first move, try second-order thinking. If the hard part is questioning your assumptions about what others want, the closed mindset guide helps.
FAQ
What are the different games in game theory?
The most common teaching examples are the Prisoner’s Dilemma, Stag Hunt, Chicken, Battle of the Sexes, Matching Pennies, and the Ultimatum game. They are a starting set. Auctions, public-goods games, and many others add more players, choices, or information.
What is the “battle of the sexes” in game theory?
It is a coordination game where both players prefer being together to being apart but prefer different shared outcomes. In the example, Alex prefers the concert and Blair prefers the match. Both shared venues are equilibria.
Is Split or Steal a prisoner’s dilemma?
It is a close variant. In the classic dilemma, defecting is strictly better whatever the other player does. In Split or Steal, stealing against a stealer pays the same zero as splitting, so Steal is only weakly dominant.
Your 5-minute game check
Pick one situation where your result depends on someone else: a shared deadline, a negotiation, a household chore.
Players:
Each player's realistic options:
Rank the outcomes for each player (best to worst):
Which classic game does that ranking match?
Pure equilibria (one-player-switch check):
Is it one-shot or repeated?
What could change the game (a rule, a signal, a commitment)?If the rankings match none of the six, trust the rankings over the label. To put a messy situation into words before you fill in the card, the Save Effort prompt library is a good place to start.