In this blog post, we’ll explore the basic concepts of game theory and how the principle of Nash equilibrium explains various social phenomena.
Game theory is a framework for analyzing situations in which two or more participants seek to maximize their own interests while taking each other’s choices into account. Such situations arise not only in sports but also in various fields such as economics, politics, business, and social life. A classic example is a penalty kick in soccer. The striker must instantly decide which direction to kick the ball, while the goalkeeper must predict this and dive accordingly. The striker may use feints to deceive the opponent, and the goalkeeper may succeed or fail in their prediction. Both players must prepare the strategy most advantageous to themselves while taking the opponent’s choice into account.
As such, rational decision-making is influenced by the surrounding environment and the opponent’s actions. In society, we sometimes cooperate with others for mutual benefit, and at other times, we compete or clash depending on our interests. The concept that helps us make the best choices amid these diverse interactions is the Nash Equilibrium in game theory. Let’s now examine the basic concepts of game theory and explore what the Nash Equilibrium is.
In a game, participants always consider their opponents’ actions when determining their own behavior. These choices are called strategies. A strategy can be a single decision, such as taking a shot in soccer, or a process involving a series of choices, such as in Go or chess. Once all participants have chosen their strategies and reach a state where no one is willing to change their strategy, an equilibrium is established. Finding an equilibrium in a game can ultimately be described as the process of identifying the most rational strategy.
However, the method for finding an equilibrium varies depending on the type of game. The most important criterion is the order in which decisions are made. Games in which players choose their strategies one after another are called sequential games, while games in which players choose their strategies simultaneously are called simultaneous games. Go and chess are classic examples of sequential games. In such games, all possible strategies available to the players can be listed in order; this is called a game tree. Players can identify the most rational strategy by analyzing the game tree in reverse order, a method known as backward induction.
In simultaneous games, a game tree cannot be used because players cannot determine their own strategy after observing their opponent’s choice. Instead, if there exists a strategy that always yields the best outcome regardless of the opponent’s choice, the player will select that strategy. Such a strategy is called a dominant strategy. Even if a dominant strategy exists for only one player, the opponent will choose their best strategy while taking this into account, allowing an equilibrium to form. However, there are many games, such as Rock-Paper-Scissors, where dominant strategies do not exist at all.
To address this issue, mathematician John Nash further generalized the concept of equilibrium. He assumed that all players repeatedly adjust their strategies while taking their opponents’ strategies into account. Each player makes the choice that is most advantageous to them in the current situation, and as this process repeats, a point is eventually reached where no one has a reason to change their strategy. The state in which each player is choosing the “best response” to the opponent’s strategy is precisely the Nash equilibrium.
A Nash equilibrium does not refer to just a single most rational strategy. A single game may have multiple Nash equilibria, and equilibria formed by dominant strategies are also included in the category of Nash equilibria.
A representative example of a Nash equilibrium is the Hawk-Dove Game, which is widely used in evolutionary biology. When animals of the same species encounter each other, there are broadly two strategies they can choose. One is the “Hawk” strategy, in which the animal fights aggressively to the end, and the other is the “Dove” strategy, in which the animal retreats when faced with an aggressive opponent. When two Doves meet, they merely threaten each other, and the conflict ends when one gives up first.
It is assumed that when two individuals employing the same strategy face off, the outcome is determined with a 50% probability. A victory yields a payoff of 100, while a loss yields no payoff. Assuming that a fight between two Hawks carries the risk of serious injury and incurs additional costs, and that a standoff between two Doves incurs the cost of wasted time, we can construct a payoff matrix for each strategy.
Analyzing this payoff matrix reveals that when one player chooses the hawk strategy and the other chooses the dove strategy, neither player is willing to change their strategy, so a Nash equilibrium is established.
If we consider a mixed strategy, in which strategies are chosen with certain probabilities, we can identify another Nash equilibrium. If the probability that the opponent chooses the hawk strategy is x, then the probability of choosing the dove strategy is 1−x. When calculating the expected payoff for each strategy, there exists a point where the expected payoffs for the hawk and dove strategies are equal.
If the opponent’s hawk strategy proportion is lower than this equilibrium point, it becomes advantageous to choose the hawk strategy; conversely, if it is higher, it becomes advantageous to increase the proportion of the dove strategy. Since the opponent will also adjust their strategy in the same way, the strategy proportions of both sides eventually converge to a constant value. In this state, neither party has a reason to change their strategy, so a new Nash equilibrium is established.
Conversely, if this mixed-strategy equilibrium is not reached, the participants’ strategies will converge back to the pure-strategy Nash equilibrium shown in the payoff matrix.
Today, game theory and the Nash equilibrium are used as key tools for analyzing rational strategies in environments where players influence one another. They are widely applied in various fields—including economics, business administration, political science, the social sciences, evolutionary biology, artificial intelligence, and computer science—and play a crucial role in understanding complex real-world decision-making.
We, too, constantly interact with others and make repeated choices in our daily lives. Decisions such as whether to cooperate or compete, or whether to compromise or stand our ground, are all situations that can be explained by game theory. If you ever find yourself wondering what decision to make at a critical moment, try thinking about game theory and the Nash equilibrium. A mindset that takes the other party’s choices into account will serve as a good guide for making more rational judgments.
