﻿ Game Theory

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# Game Theory

## Definition

Game theory - a mathematical method of decision making in which alternative strategies are analysed to determine the optimal course of action for the interested party, depending on assumptions about rivals’ behaviour. Widely used in economics, game theory is also used as a tool in biology, psychology and politics.

## Game theory map

There are to forms of game theory: 1) strategic form 2) extensive form . 1) The strategic form covers Mixed strategies, Prisoners’ dilemma and incomplete information. Within this, the prisoners’ dilemma touches on rational behaviour and enforcement; whereas enforcement involves cooperation.

2)

## Strategic form and Extensive form game theory

A strategic form is the right choice for simultaneous games whereby both players make their move simultaneously given the information which they have available. A mixed strategy can be seen when there are two players who will not agree on an equilibrium, meaning that a mixed strategy may be adopted which should end with a balance, essentially maximising the utility based on the different opinions. The other option to take was the extensive form. An extensive form game could be visualised as a tree, presenting every possible state of play which can be played in the game.

## Prisoner’s dilemma

If none of them confesses (they cooperate with each other) they will both be charged the lesser sentence, a year each in prison. The police will question them on separate interrogation rooms, which means that the two prisoners cannot communicate (hence, imperfect information). The police will try to convince each prisoner to confess the crime by offering them a “get out of jail free card,” while the other prisoner will be sentenced to a ten years term. If both prisoners confess (and therefore they defect), each prisoner will be sentenced to eight years. Both prisoners are offered the same deal and know the consequences of each action (complete information) and are completely aware that the other prisoner has been offered the exact same deal (therefore, there is common knowledge).

Brief description of the strategy

Ultimately, the ‘Nash Equilibrium ‘may be for each to remain silent, shortening each of their sentences. However, when considering a dominant strategy, it may be considered that the iterated dominance would be for each person to speak out, in turn pushing the other player to also speak. Essentially, iterated dominance is seen as the dominant strategy for that individual player, in the case above being for the player to speak out, and potentially receive 0 years. The main difference to these two options is the level of knowledge, or collusion that exists between the parties. As mentioned with the case above, there is no interaction, and so each player can only predict the movement of the other player, rather than know for certain.

Iterated dominance

A normal prisoner's dilemma played repeatedly by the same participants. An iterated prisoner's dilemma differs from the original concept of a prisoner's dilemma because participants can learn about the behavioural tendencies of their counterparty. This is better explained using an example as can be seen below:

the choice by two businesses to invest \$100Million to expand production.

 X Plc Y Plc Invest 50% Investment Don’t Invest Invest 5, 5 10, 4 25, -10 Don’t Invest -10, 25 -8, 10 0,0

this game could potentially be solved by the iterated deletion of dominant strategies. The main idea here is that by understanding the dominant strategies, a prediction can be made into the choice of the business. Iterated dominance is built on two main assumptions.

Assumptions:

1. even if the player doesn’t have a dominant strategy, they may still have one which dominates another, for instance, by investing, Y Plc., has a higher payoff in all instances. So, what is being mentioned here is that, instead of using a superlative statement (i.e. Investing in the best option available), a comparative statement can be used; namely investing would be a better option than not investing. For Y Plc., given that the payoff is higher in all instances with investing, we can say that the ‘Invest’ strategy dominants the ‘Don’t Invest’ strategy.
2. The second idea is then over the transition from some dominant strategies to iterated dominance; seen as anticipating the moves of the other player. To simplify this, consider that Y Plc., benefits in all instances if it chooses to ‘Invest’. With this, X Plc., should recognised that investing is a strictly dominant strategy for Y Plc., and so in this case there is no chance that Y Plc., will consider the ‘Don’t Invest’ strategy. Ultimately, it is a rational decision for Y Plc., to invest under all circumstances. So, by eliminating the ‘Don’t Invest’ move, X Plc., reducing the number of outcomes, helping make its own decision.

Nash equilibrium

Definition: a concept of (a non-cooperative game) game theory where the optimal outcome of the game is where no player has an incentive to deviate from their chosen strategy, even after considering their opponent’s choice. Overall, the individual can receive no incremental benefit from changing actions; under the assumption that all other players keep their strategy constant. A game may have multiple Nash Equilibria or none at all.

Stated simply, Alice and Bob are in Nash equilibrium if Alice is making the best decision she can, taking into account Bob's decision while Bob's decision remains unchanged, and Bob is making the best decision he can, taking into account Alice's decision while Alice's decision remains unchanged. Likewise, a group of players are in Nash equilibrium if each one is making the best decision possible, taking into account the decisions of the others in the game as long as the other parties' decisions remain unchanged.

Game theorists use the Nash equilibrium concept to analyze the outcome of the strategic interaction of several decision makers. In other words, it provides a way of predicting what will happen if several people or several institutions are making decisions at the same time, and if the outcome depends on the decisions of the others. The simple insight underlying John Nash's idea is that one cannot predict the result of the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do, taking into account the decision-making of the others.

Nash equilibrium has been used to analyze hostile situations like war and arms races[2] (see prisoner's dilemma), and also how conflict may be mitigated by repeated interaction (see tit-for-tat). It has also been used to study to what extent people with different preferences can cooperate (see battle of the sexes), and whether they will take risks to achieve a cooperative outcome (see stag hunt). It has been used to study the adoption of technical standards,[citation needed] and also the occurrence of bank runs and currency crises (see coordination game). Other applications include traffic flow (see Wardrop's principle), how to organize auctions (see auction theory), the outcome of efforts exerted by multiple parties in the education process,[3] regulatory legislation such as environmental regulations (see tragedy of the Commons),[4] natural resource management,[5] analysing strategies in marketing,[6] and even penalty kicks in football (see matching pennies)

Different types of equilibrium -

1. Zero sum games - the gain of one player is offset by a loss from another. For instance, consider two businesses operating in a market worth £100Million. Business B chooses to spend 10% more on its marketing, while Business A chooses to remain the same. The result would be that Business B gains £9Million in sales, while Business A loses £9Million sales; ultimately the game is ‘zero sum’ as the gains posted by Business B are the losses posted by Business A.
2. Non-cooperative games - games where players make their decisions independently. Thus, while players could cooperate, any cooperation must be self-enforcing. good example of this could be considered in the oil market with OPEC and non-OPEC. The current strategy adopted calls for production from all OPEC members to be capped at 32Million bpd, a target which is enforced by all for members for the benefit of the market. However, what the OPEC example also shows us is that for the benefits to be realised, cooperation would need to include all in the market. So, while OPEC have cooperated to cut production, higher prices in the short-term have only enticed non-OPEC producers to increase their own production, leading to price falls; keeping oil prices low. Non-cooperative games can become difficult to consider given that there is also the need to take into account bargaining power of the players.
3. Multi-move games - When it comes to mentioning game theory in the business world, it becomes hard to imagine a game where there is only one move. When we imagine businesses games with multiple players, each may take multiple moves over several periods/ years; in turn competing with each other and responding to changes in the marketplace. In game theory, a sequential game is where one player chooses their move before the other. With this in mind, the expectation would be that after the first player has chosen, player 2 has a greater knowledge, and so could be better informed into how to respond. The game can be visualised in the game tree shown below:

Visualisation of a ‘Game Tree’ - these can get bigger depending on the number of decisions which can be made by the players. There could also be more than two players involved, as we see with many competitive, and dynamic markets.

The importance of knowledge, noting how players can make more informed choices in sequential games given the increase in information which is available.

1. Single/ multi-player games - Following on from the above, we can also consider games which have multiple players. While game theory was originally developed on the back of duopoly behaviour in markets; changes now mean there are many more competitors, and so players within the market. For instance, consider a flight between London - Tokyo. Multiple players can have a position in the market, be it the direct operators such as British Airways, other ‘hub’ airlines such as Emirates and even the low-cost carriers such as EasyJet who may offer connections in other European cities. This creates a game whereby there are more than two players, creating the potential for many more combinations of outcomes.

For instance, consider the construction of a rota in a 24/7 retail outlet with 100 employees. The payoff associated with different working hours may differ based on the pay, and the utility gained/ lost from working the shift. Essentially, this is a game involving 100 players who will all have their own strategies based on individual preferences.