Strategy (game theory)

Imagine playing a game like chess or poker. Your choice depends on what others do. A strategy is a full plan for every turn. It is like a list of directions. You use this list to help you win. Sometimes, you might pick a move by chance. This keeps your plan a secret. This way, others cannot guess your next move. Using a good plan helps you play better games.

Imagine you are playing a game like chess or poker. Your success depends on what you do. It also depends on what others do. This is the heart of game theory.

A strategy is a complete plan. It is a list of directions for every turn. It tells you what to do in any situation. You can have a pure strategy. This is a fixed plan that never changes. You might also use a mixed strategy. This means you pick your moves by chance. Using chance can help you stay a secret. If others cannot guess your move, they cannot stop you.

In a soccer penalty kick, players use these ideas. The kicker chooses a side to shoot. The goalie chooses a side to block. If the goalie guesses right, the kick is blocked. If the goalie guesses wrong, the ball might go in. Because players want to win, they must balance their choices.

A man named John Forbes Nash studied these plans. He proved that every finite game has an equilibrium. An equilibrium is a state where no player wants to change their plan. Some games use pure strategies to reach this state. Other games need mixed strategies to find it.

Imagine you are playing a game like chess or poker. Your success depends on your own choices. It also depends on what other people choose to do. This is the core idea of game theory. A strategy is much more than just a single move. It is a complete list of directions for every possible situation. You can think of a strategy as a set of rules. These rules tell you exactly what to do at every turn. This helps you plan for what might happen next.

There are different ways to follow a plan. A pure strategy is a fixed and certain plan. It tells you exactly what action to take every single time. A mixed strategy is different because it uses chance. Instead of one fixed plan, you pick between different plans using probabilities. This can be very helpful in games. If you are predictable, other players can stop you. By using a mixed strategy, you stay a secret.

We can see these ideas in a soccer penalty kick. The kicker chooses to shoot left or right. At the same time, the goalie chooses to lean left or right. If the goalie guesses the right way, the kick is blocked. A study by Chiappori, Levitt, and Groseclose in 2002 looked at this. They found that professional players do use mixed strategies in real life. Kickers often shoot to their favorite side about 45 percent of the time. Goalies lean toward that same side about 57 percent of the time.

A famous mathematician named John Forbes Nash studied these patterns. He proved that every finite game has an equilibrium. An equilibrium is a special state in a game. It is a point where no player wants to change their plan. Some games reach this state using pure strategies. Other games, like the game of matching pennies, need mixed strategies. These mixed strategies help players find the best balance.

Game theorists also look at how much information players have. In a Bayesian game, players might not know everything about each other. This means their strategy set includes rules for any private information. There are also games played over time, called dynamic games. In these games, a strategy might be a set of rules for a robot. Even the way we think about these plans is changing. Some experts see mixed strategies as a way to show what we do not know. Others see them as a way to describe how large groups of people act.

In the field of game theory, a strategy is a complete plan for playing a game. This differs from a single move or action. A move is just one choice made during one turn. In contrast, a strategy is like a complete algorithm or a list of directions. It tells a player exactly what to do for every possible situation that might arise. The goal of a player is to maximize their payoff, which is the outcome or reward they receive. Because the success of one player depends on the actions of others, players must consider how competitors will behave.

To find the best path, a player analyzes different scenarios. For example, if Competitor A is deciding whether to enter a market, they first assume Competitor B will enter. Competitor A then compares the payoffs for entering versus not entering. Next, they assume Competitor B will not enter and re-evaluate their options. This process can help identify a dominant strategy. A dominant strategy is an action that provides the best payoff regardless of what the competitor chooses to do.

Every game has a strategy set, which defines all the possible strategies available to a player. These sets can be finite or infinite. A finite strategy set contains a specific, countable number of discrete options. In the game of rock paper scissors, each player has a finite strategy set consisting of {rock, paper, scissors}. An infinite strategy set occurs when there are endless possibilities. In a cake-cutting game, a player could choose to cut at any point between zero and 100 percent of the cake. This creates a continuous range of choices.

Strategies are often categorized as either pure or mixed. A pure strategy is a deterministic plan. It specifies exactly one action for every decision point, given any information available. A mixed strategy is a probability distribution over the set of pure strategies. Instead of committing to one plan, a player randomizes between different pure strategies using specific probabilities. This is useful in games where no single pure strategy is a best response. By using a mixed strategy, players can avoid being predictable. A totally mixed strategy is one where every possible pure strategy is assigned a positive probability.

A practical example of mixed strategies is a soccer penalty kick. The kicker chooses to shoot left or right, while the goalie chooses to lean left or right. If the goalie guesses correctly, the kick is blocked, resulting in a payoff of zero for both. If the goalie guesses wrong, the kicker receives a higher payoff for shooting to their preferred side. A 2002 study by Chiappori, Levitt, and Groseclose examined this in professional play. They found that kickers shoot to their favored side 45 percent of the time. Goalies lean toward that same side 57 percent of the time.

The mathematician John Forbes Nash made a major breakthrough in this field. He proved that every finite game has at least one Nash equilibrium. An equilibrium is a state where no player can improve their payoff by changing their strategy alone. Some games reach equilibrium through pure strategies, such as the Prisoner's Dilemma or the Stag Hunt. Other games, like matching pennies, require mixed strategy Nash equilibria. In these cases, at least one player must use a randomized plan to maintain the balance.

There are different ways to interpret why players use mixed strategies. In the 1970s, Harsanyi proposed the idea of purification. This suggests that mixed strategies simply reflect our lack of knowledge about a player's exact decision process. Another view sees mixed strategies as representing a large population of agents. In this model, the strategy describes the fraction of the population choosing each option. Finally, some theorists view equilibrium as a state of beliefs. In a game like rock paper scissors, an equilibrium in beliefs means players believe their opponents are equally likely to play any move.