Нөлдік қосым ойыны

Situation where total gains match total losses

Zero sum game is a mathematical representation in game theory and economic theory of a situation that involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is equivalent to player two's loss, with the result that the net improvement in benefit of the game is zero. If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a more significant piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero sum game if all participants value each unit of cake equally. Other examples of zero sum games in daily life include games like poker, chess, and bridge where one person gains and another person loses, which results in a zero net benefit for every player. In the markets and financial instruments, futures contracts and options are zero sum games as well. In contrast, non zero sum describes a situation in which the interacting parties' aggregate gains and losses can be less than or more than zero. A zero sum game is also called a strictly competitive game, while non zero sum games can be either competitive or non competitive. Zero sum games are most often solved with the minimax theorem which is closely related to linear programming duality,

The zero sum property (if one gains, another loses) means that any result of a zero sum situation is Pareto optimal. Generally, any game where all strategies are Pareto optimal is called a conflict game. Zero sum games are a specific example of constant sum games where the sum of each outcome is always zero. Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation. In situation where one decision maker's gain (or loss) does not necessarily result in the other decision makers' loss (or gain), they are referred to as non zero sum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non zero sum situation. Other non zero sum games are games in which the sum of gains and losses by the players is sometimes more or less than what they began with. The idea of Pareto optimal payoff in a zero sum game gives rise to a generalized relative selfish rationality standard, the punishing the opponent standard, where both players always seek to minimize the opponent's payoff at a favourable cost to themselves rather than prefer more over less. The punishing the opponent standard can be used in both zero sum games (e. g. warfare game, chess) and non zero sum games (e. g. pooling selection games). The player in the game has a simple enough desire to maximise the profit for them, and the opponent wishes to minimise it.

For two player finite zero sum games, the different game theoretic solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. If the players are allowed to play a mixed strategy, the game always has an equilibrium.

The Nash equilibrium for a two player, zero sum game can be found by solving a linear programming problem. Suppose a zero sum game has a payoff matrix M where element is the payoff obtained when the minimizing player chooses pure strategy i and the maximizing player chooses pure strategy j (i. e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of M is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (Raghavan 1994, p. 740) by solving the following linear program to find a vector u:

The first constraint says each element of the u vector must be nonnegative, and the second constraint says each element of the M u vector must be at least 1. For the resulting u vector, the inverse of the sum of its elements is the value of the game. Multiplying u by that value gives a probability vector, giving the probability that the maximizing player will choose each possible pure strategy. If the game matrix does not have all positive elements, add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will not affect the equilibrium mixed strategies for the equilibrium. The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Alternatively, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of M (adding a constant so it is positive), then solving the resulting game. If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two player, zero sum game by using a change of variables that puts it in the form of the above equations and thus such games are equivalent to linear programs, in general.

If avoiding a zero sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero sum game. For any two players zero sum game where a zero zero draw is impossible or non credible after the play is started, such as poker, there is no Nash equilibrium strategy other than avoiding the play. Even if there is a credible zero zero draw after a zero sum game is started, it is not better than the avoiding strategy. In this sense, it's interesting to find reward as you go in optimal choice computation shall prevail over all two players zero sum games concerning starting the game or not. The most common or simple example from the subfield of social psychology is the concept of "social traps". In some cases pursuing individual personal interest can enhance the collective well being of the group, but in other situations, all parties pursuing personal interest results in mutually destructive behaviour. Copeland's review notes that an n player non zero sum game can be converted into an (n+1) player zero sum game, where the n+1st player, denoted the fictitious player, receives the negative of the sum of the gains of the other n players (the global gain / loss).

It is clear that there are manifold relationships between players in a zero sum three person game, in a zero sum two person game, anything one player wins is necessarily lost by the other and vice versa; therefore, there is always an absolute antagonism of interests, and that is similar in the three person game. A particular move of a player in a zero sum three person game would be assumed to be clearly beneficial to him and may disbenefits to both other players, or benefits to one and disbenefits to the other opponent.====

Studies show that the entry of low cost airlines into the Hong Kong market brought in $671 million in revenue and resulted in an outflow of $294 million. Therefore, the replacement effect should be considered when introducing a new model, which will lead to economic leakage and injection. Thus introducing new models requires caution. For example, if the number of new airlines departing from and arriving at the airport is the same, the economic contribution to the host city may be a zero sum game. Because for Hong Kong, the consumption of overseas tourists in Hong Kong is income, while the consumption of Hong Kong residents in opposite cities is outflow. In addition, the introduction of new airlines can also have a negative impact on existing airlines. Consequently, when a new aviation model is introduced, feasibility tests need to be carried out in all aspects, taking into account the economic inflow and outflow and displacement effects caused by the model.

Derivatives trading may be considered a zero sum game, as each dollar gained by one party in a transaction must be lost by the other, hence yielding a net transfer of wealth of zero. An options contract whereby a buyer purchases a derivative contract which provides them with the right to buy an underlying asset from a seller at a specified strike price before a specified expiration date – is an example of a zero sum game. A futures contract – whereby a buyer purchases a derivative contract to buy an underlying asset from the seller for a specified price on a specified date – is also an example of a zero sum game. This is because the fundamental principle of these contracts is that they are agreements between two parties, and any gain made by one party must be matched by a loss sustained by the other. If the price of the underlying asset increases before the expiration date the buyer may exercise/ close the options/ futures contract. The buyers gain and corresponding sellers loss will be the difference between the strike price and value of the underlying asset at that time. Hence, the net transfer of wealth is zero. Swaps, which involve the exchange of cash flows from two different financial instruments, are also considered a zero sum game. Consider a standard interest rate swap whereby Firm A pays a fixed rate and receives a floating rate; correspondingly Firm B pays a floating rate and receives a fixed rate. If rates increase, then Firm A will gain, and Firm B will lose by the rate differential (floating rate – fixed rate). If rates decrease, then Firm A will lose, and Firm B will gain by the rate differential (fixed rate – floating rate). Whilst derivatives trading may be considered a zero sum game, it is important to remember that this is not an absolute truth. The financial markets are complex and multifaceted, with a range of participants engaging in a variety of activities. While some trades may result in a simple transfer of wealth from one party to another, the market as a whole is not purely competitive, and many transactions serve important economic functions. The stock market is an excellent example of a positive sum game, often erroneously labelled as a zero sum game. This is a zero sum fallacy: the perception that one trader in the stock market may only increase the value of their holdings if another trader decreases their holdings. The primary goal of the stock market is to match buyers and sellers, but the prevailing price is the one which equilibrates supply and demand. Stock prices generally move according to changes in future expectations, such as acquisition announcements, upside earnings surprises, or improved guidance. For instance, if Company C announces a deal to acquire Company D, and investors believe that the acquisition will result in synergies and hence increased profitability for Company C, there will be an increased demand for Company C stock. In this scenario, all existing holders of Company C stock will enjoy gains without incurring any corresponding measurable losses to other players. Furthermore, in the long run, the stock market is a positive sum game. As economic growth occurs, demand increases, output increases, companies grow, and company valuations increase, leading to value creation and wealth addition in the market.

It has been theorized by Robert Wright in his book Nonzero: The Logic of Human Destiny, that society becomes increasingly non zero sum as it becomes more complex, specialized, and interdependent.

In 1944, John von Neumann and Oskar Morgenstern proved that any non zero sum game for n players is equivalent to a zero sum game with n + 1 players; the (n + 1)th player representing the global profit or loss.

Zero sum games and particularly their solutions are commonly misunderstood by critics of game theory, usually with respect to the independence and rationality of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational games. Politics is sometimes called zero sum because in common usage the idea of a stalemate is perceived to be "zero sum"; politics and macroeconomics are not zero sum games, however, because they do not constitute conserved systems.

In psychology, zero sum thinking refers to the perception that a given situation is like a zero sum game, where one person's gain is equal to another person's loss.