A game may have many Nash equilibria, or none. The Brouwer fixed point theorem provides the sufficient, though not necessary, conditions for existence of a Nash equilibrium. In a game context, if the set of strategies by player i, is a compact and continuous set, the payoff functions for all players are quasi-concave and continuous, then the game has a Nash Equilibrium NE.
A game can have a pure strategy NE or a NE in its mixed extension of choosing a pure strategy stochastically with a fixed frequency.here
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Nash was able to prove that, if we allow mixed strategies players choose strategies randomly according to pre-assigned probabilities , then every n -player game in which every player can choose from finitely many strategies admits at least one Nash equilibrium. Consider the following two-player game: both players simultaneously choose a whole number from 0 to Both players then win the minimum of the two numbers in dollars.
This game has a unique Nash equilibrium: both players have to choose 0. Any other choice of strategies can be improved if one of the players lowers his number to one less than the other player's number.
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If the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 11 Nash equilibria. Main article: Coordination game. The coordination game is a classic symmetric two player, two strategy game, with payoff matrix. The players should thus cooperate on either of the two strategies to receive a high payoff.
Players in the game have to agree on one of the two strategies in order to receive a high payoff. If the players do not agree, a lower payoff is rewarded. An example of a coordination game is the setting where two technologies are available to two firms with compatible products, and they have to elect a strategy to become the market standard.
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- Existence of Nash equilibria.
If both firms agree on the chosen technology, high sales are expected for both firms. If the firms do not agree on the standard technology, few sales result. Both strategies are Nash equilibria of the game. Driving on a road, and having to choose either to drive on the left or to drive on the right of the road, is also a coordination game.
For example, with payoffs meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:.
In this case there are two pure strategy Nash equilibria, when both choose to either drive on the left or on the right. Main article: Prisoner's dilemma.
The Prisoner's dilemma has one Nash equilibrium: when both players defect. However, "both defect" is inferior to "both cooperate", in the sense that the total jail time served by the two prisoners is greater if both defect. The strategy "both cooperate" is unstable, as a player could do better by defecting while their opponent still cooperates. Thus, "both cooperate" is not an equilibrium. As Ian Stewart put it, "sometimes rational decisions aren't sensible! The concept of stability , useful in the analysis of many kinds of equilibrium can also be applied to Nash equilibria. A Nash equilibrium for a mixed strategy game is stable if a small change specifically a infinitesimal change in probabilities for one player leads to a situation where two conditions hold:.
One of the fathers of game theory , died together with his wife in a car accident last weekend.
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John Forbes Nash, aged 86, had just came back from Oslo where he collected the prestigious Abel Prize from the Norwegian Academy of Science and Letters for his mathematical work. Interestingly, 21 years ago now he also came back from Scandinavia with another prize. On that occasion he was awarded the Nobel Prize in Economics for his pioneering mathematical work in the s.
The prize recognized his work on the concepts of equilibrium in game theory.
His analyses in cooperative and non-cooperative games, known as the Nash Equilibrium , provide solutions to situations in which two or more parties interact to make decisions. His work have played since an essential role in the development of many branches of science, such as politics, biology, philosophy, computer science and of course economics.
To exchange bitcoins and fiat money in Coinffeine both counterparties are matched automatically, without knowing who your counterparty is. This could have big risks if not for beautiful minds like Nash who have made this possible. With the death of Nash, the world lost a genius.
But fortunately his legacy will remain and the applications it may have, will be immortal. The next concept that interests us is mutual assured destruction , a concept closely related to the years of the Cold War. To get some atmosphere going, we can think of the cult film classic: Dr. At that time nuclear power was already developed, and had sadly been used as a closing ceremony of the Second World War. Both powers had entered into an arms race which, if continued, could have ended with humanity on several occasions. For years it seemed that nuclear war was about to happen anytime, but it never did.
The answer to this is the fact that both players had the ability to respond to a nuclear attack, even if it was a last resort and losses would be dramatic on both sides. The advantage of being the first to attack was insignificant compared to the losses it would cause. And it justified the cost of always having intercontinental missiles ready, and nuclear submarines or planes circling the Arctic to be able to respond to an attack.
If we take stock of the possible outcomes in a table, we can see that the best answer to not attack is not attack, and in the event that the other player attacks it is indifferent what we do. In this case, we need the next best thing, a Nash equilibrium and yes, there is a film about it.
These equilibrium situations occur when each player can not improve his situation by changing his strategy if the other player persists.
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