What makes a situation a game
Most of the probability topics on this site deal with a single decision against chance: a die, a shuffled deck, a loot table. Dice do not react to what you do. Other players do. Game theory is the study of decisions where the outcome depends on the choices of several decision makers, each of whom is also trying to do well by their own standards.
That shift changes the question you ask. Against a die you ask, “What is likely to happen?” Against a rival you ask, “What will they do, given that they are asking the same question about me?” The answer can be surprising, because a choice that looks good in isolation can look poor once you account for how others respond.
Formal game theory uses a few plain building blocks:
- Players: the decision makers, whether two or ten.
- Strategies: the complete plans each player could follow, not just a single move.
- Payoffs: a number for each player at each possible outcome, standing for how much they value it. A payoff might be points, a share of a prize, or simply a ranking of outcomes from best to worst.
- Information: what each player knows when they choose, including what they know about the other players’ payoffs and earlier moves.
None of this requires advanced mathematics. It requires clarity about who wants what, and who knows what.
Payoffs and rational choice
Game theory usually assumes players are rational in a narrow sense: each picks the option that best serves their own payoffs, given what they expect others to do. That does not mean players are selfless, calm or good at arithmetic. It also does not mean they only care about money or points. The payoffs are whatever the player actually cares about, so a player who enjoys a dramatic bluff can have that built into their numbers.
Real people often depart from this idealised picture. Habits, emotions and the kinds of errors described under cognitive biases all push behaviour away from the textbook. That is not a reason to discard the theory. It is a reason to treat it as a clear reference point: a way to ask what a careful player would do, and then to ask how far actual opponents are likely to be from that.
A first example
Suppose two players each secretly choose either “Attack” or “Wait”, then reveal at the same time. The table lists the payoffs as (you, rival).
| Rival attacks | Rival waits | |
|---|---|---|
| You attack | (1, 1) | (4, 0) |
| You wait | (0, 4) | (3, 3) |
If your rival attacks, attacking gives you 1 and waiting gives you 0, so attacking is better. If your rival waits, attacking gives you 4 and waiting gives you 3, so attacking is again better. Attack is better whatever the rival does, and the same reasoning applies to them. Both attack and each receives 1, even though both waiting would have given each of them 3.
This structure, where individually sensible choices lead to a worse shared result, is the heart of the prisoner dilemma. Notice that the analysis never required guessing your rival’s mood. It only required comparing columns.
Simultaneous and sequential moves
Games differ in timing. In a simultaneous game, such as the table above, players choose without seeing the other’s choice. In a sequential game, players move in turn and can observe earlier moves, as in chess or most board games. Timing matters because observing a move is a form of information.
Sequential games are often drawn as branching diagrams called game trees, and solved from the end backward. Simultaneous games are usually shown as payoff tables. Many real games mix the two: players choose secretly in one phase and then react to what was revealed in the next.
Where the key ideas fit
A handful of concepts recur throughout this part of the site:
- A dominant strategy is a choice that does at least as well as every alternative whatever others do. When one exists, the decision becomes easy.
- A Nash equilibrium is a combination of strategies in which no player can improve their payoff by changing only their own choice. It describes a stable outcome, not necessarily a good one.
- Mixed strategies involve choosing among options with set probabilities. They matter when any predictable choice can be exploited.
- Whether a game is among the zero-sum games or is cooperative decides whether the players’ interests are opposed, aligned, or a mix.
Each of these has its own article in this cluster, so here the point is simply how they connect: dominance is the simplest tool, equilibrium is the general one, and randomisation is what you need when no stable pure choice exists.
Hidden information and signals
In many games the key unknown is not the rival’s move but the rival’s private information, such as the cards in their hand. Then play becomes a form of communication, whether you intend it or not. Every action tells observers something, and a thoughtful rival will update their beliefs accordingly, much as described in material on Bayes rule. This is why a poker player who bets only with strong hands becomes easy to read, and why mixing in some bluffs can matter. The same thinking explains why a player who studies opponent reading tries to learn what each choice reveals.
What game theory can and cannot do
Used well, game theory helps you:
- notice when your best move depends on someone else’s, and when it does not;
- remove options that no sensible player would choose;
- see why a rival might behave in a seemingly odd but sensible way;
- recognise situations where cooperation is fragile.
It has limits. Few real games have fully specified payoffs, and the number of possible strategies in something like chess is far too large to enumerate. Opponents may not be rational in the narrow sense, and an equilibrium prediction can miss the mark against someone who plays differently. Game theory also does not remove luck. Outcomes in games with dice or shuffled cards still carry uncertainty, so a sound decision can lose and an unsound one can win in any single game. A good habit is to combine it with an estimate of expected value for the chance elements and then reason about the human ones.
A practical way to start
When you meet a new game, try this short routine:
- List the real choices available to each player at each step.
- Write down roughly what each outcome is worth to each player.
- Check for options that are always worse than another, and set them aside.
- Ask what your rivals would do in your position, and what your move would tell them.
- If you keep being outguessed, consider whether more variety in your play would help.
Treat the result as a reasoned starting point rather than a guarantee. The aim is better decisions, not certainty, and you can keep refining your view of rivals as you see how they actually play.