Game theory is a small branch of mathematics, and it is also a useful subject in economics, so as to have a systematic understanding of the subject.

## Perfect Information

In economics, perfect information (sometimes referred to as “no hidden information”) is a feature of perfect competition. With perfect information in a market, all consumers and producers have complete and instantaneous knowledge of all market prices, their own utility, and own cost functions.

In game theory, a sequential game has perfect information if each player, when making any decision, is perfectly informed of all the events that have previously occurred, including the “initialization event” of the game (e.g. the starting hands of each player in a card game).

Perfect information is importantly different from complete information, which implies common knowledge of each player’s utility functions, payoffs, strategies and “types”. A game with perfect information may or may not have complete information.

Games where some aspect of play is hidden from opponents – such as the cards in poker and bridge – are examples of games with imperfect information.

## Complete Information

In economics and game theory, complete information is an economic situation or game in which knowledge about other market participants or players is available to all participants. The utility functions (including risk aversion), payoffs, strategies and “types” of players are thus common knowledge. Complete information is the concept that each player in the game is aware of the sequence, strategies, and payoffs throughout gameplay. Given this information, the players have the ability to plan accordingly based on the information to maximize their own strategies and utility at the end of the game.

## Incomplete Information

Inversely, in a game with incomplete information, players do not possess full information about their opponents. Some players possess private information, a fact that the others should take into account when forming expectations about how those players will behave. A typical example is an auction: each player knows his own utility function (valuation for the item), but does not know the utility function of the other players.

## Decision Tree

Whether you play chess, go, backgammon or other decision-making games, you can draw a decision tree, they are essentially different in rules and complexity, because their decision-making steps are limited, theoretically , if we can traverse all the situations in the decision tree, we can know the result of the game, win or lose or tie.

Game Complexity can be measured in many ways:

- State-space Complexity
- Game-tree Complexity
- Game-tree Size
- Decision Complexity
- Computational Complexity
- …

## Zermelo’s Theorem

In game theory, Zermelo’s theorem is a theorem about finite two-person games of perfect information in which the players move alternately and in which chance does not affect the decision making process. It says that if the game cannot end in a draw, then one of the two players must have a winning strategy (i.e. can force a win). An alternate statement is that for a game meeting all of these conditions except the condition that a draw is now possible, then either the first-player can force a win, or the second-player can force a win, or both players can at least force a draw.

The theorem is named after Ernst Zermelo, a German mathematician and logician, who proved the theorem for the example game of chess in 1913.

It tells us that, assuming that both sides are chess masters and know the game well, they will definitely adopt a unified strategy at this time to make the game develop in a fixed direction, and the final outcome is also fixed. In the end, it will definitely be a draw, because Any party that unilaterally changes its decision will be detrimental to itself.

## Nash Equilibrium

In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players.

In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one’s own strategy.

It tells us, as long as the strategies of several parties participating in the game are limited, there must be an equilibrium state, and everyone will adopt this balanced strategy, and there is no motivation to unilaterally change the strategy.

## Classic Examples

In general, the optimal strategy is a Nash equilibrium, and Zermelo’s Theorem is an example of a Nash equilibrium. And the world we live in is full of examples of Nash equilibrium no matter what aspects.

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