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Game Theory: A Brief Introduction

8 November 2018

Game Theory

 

the analysis of strategies for dealing with competitive situations between two or more players

All I do is win

In the many games of life the old saying goes “Sometimes you win sometimes you lose”. What if you could always win, or at best avoid losing? This same question was posed by mathematician and Nobel prize winner John Nash. Answering it bloomed a new field of mathematics known as game theory. Game theory is the analysis of strategies for dealing with competitive situations between two or more players. The math’s behind this theory are applicable to a wide range of topics, such as describing human behavior or explaining biological evolution. Game theory can be applied to a wide array of interactions, hence it’s important to understand the different strategies that emerge and use them to achieve an optimal outcome.

Nash Equilibrium

 

a stable state, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged.

Befriending Nash

Prisoners Dilemma

So game theory seems cool and all but how can I apply it? Well, for starters one must understand the game case known as Prisoners Dilemma. Say for example that a random company has been charged with fraud and its two top executives are brought into interrogation. These executives could now consider to be in a ‘game situation’. Say player one is named Jeff and player two is Andy (if you’re thinking of Enron execs this is pure coincidence). The judge is lacking some evidence and can only convict them to 10 years in prison. However, in the separate interrogation rooms each player is offered the opportunity to snitch on his colleague and reduce his own sentence to 5 years. In contrast by offering incriminating evidence, his coworker’s sentence can be increased. If one stays quiet while the other snitches all the blame can go to one player giving him 25 years in prison. If both snitch then each receive a 20 year sentence. Each “player” is now confronted with two choices he can either stay quiet or incriminate his coworker. Each choice can have a different payoff depending on what the OTHER player decides. This diagram summarizes all possibilities.

Andy Stays Quiet Andy Incriminates Jeff
Jeff Stays Quiet Andy Serves:  10 years

Jeff Serves:    10 years

Andy Serves:  5 years

Jeff Serves:    25 years

Jeff Incriminates Andy Andy Serves:   25 years

Jeff Serves:       5 years

Andy Serves:  20 years

Jeff Serves:    20 years

Two interesting scenarios result. The first one is where both stay quiet and each serve 10 years (top left). This is understood as the most efficient outcome for the team (Pareto efficient outcome). The second is where both incriminate each other and serve each 20 years (bottom right). This is understood as the Nash Equilibrium. It’s the best option for the individual player irrelevant of the other player’s action. In other words, if Jeff had not agreed with Andy to stay quiet his best option as an INDIVIDUAL is to incriminate Andy. At best he would only serve 5 years and at worst he would serve 20 years. This is still less than the top right scenario where he would serve 25 years if he keeps quiet and Andy snitches on him.

The key insight here is that the best solution for the GROUP is for both to keep quiet. The pareto efficient outcome (10 year each) is only guaranteed if the both players decide to keep quiet. However, an individual player has strong incentive to betray his friend and opt to get the 5 year sentence. In the long term both players tend to betray each other and consequently receive each a 20 year sentence. This outcome is clearly worse for both. Had they agreed to both keep quiet, they would have both served half the time in prison.

 

 

 

 

 

Master Strategy

Andy Fastow (Former Enron CFO) served 6 years in prison for fraud, and his coworker Jeff Skilling (Former Enron CEO) completed this August his 11 year prison sentence. Andy Fastow’s prison time was reduced from the original 10 years to 6 years after cooperating with the prosecution of former Enron employees.

So, what? These Enron guys ended up losing and I thought this article was supposed to teach me how to win. Well to win one must first understand the different games and strategies. The key is in the payoffs of each player since this influence’s individual strategy. If you can change the payoffs, you can change your opponent’s strategies, you can change the game.

Take for example Samsung and Apple as two players in the game to sell you a phone. Both companies would like to sell phones at the highest price the market is willing to pay, but both companies have to compete with each other to reach that market. Say for example, that both companies equally share the market with 50% buying Apple and other 50 % buying Samsung. They both have the option to keep their current price or lower their price. By lowering their price to their competitor’s, they may steal their customers and gain larger profits. Here is the payoffs table describing the profits of each company after keeping or lowering their price.

Samsung keeps price Samsung lowers price
Apple keeps price Samsung Profits:     50$

Apple Profits:          50$

Samsung Profits :     70$

Apple  Profits:           30$  

Apple lowers price Samsung Profits:     30$

Apple Profits:          70$

Samsung Profits :      30$

Apple Profits:            30$

This game’s payoffs are identical to that of the prisoner’s dilemma. The scenario where both companies would act in their own interest to make more money would lead both to lower each other’s price. The resulting Nash Equilibrium is where both Apple and Samsung reduce their prices. However, both Apple and Samsung could achieve larger profits if both of them got together agreed to keep the price of their phones (top left). The buyer would have no option than to spend more money on his phone irrelevant of the company he chooses. This is known as price fixing. Besides not being fair to consumers such “price monopoly” is completely illegal. This however is the pareto optimal outcome in which both companies end up winning.

With the temptation for a company to lower its phone price to attract more customers how can both companies ensure the pareto outcome (top left)?

Here is where understanding each companies’ payoffs and tweaking them can result in a mutually optimal outcome. Winning (dab). Say Apple spent an expensive marketing campaign of their new iPhone Xs selling price (only 1199 CHF). If Apple then wanted to change its selling price the cost for a new advertisement campaign would lower its profit. This is shown in the table below by a -30$. This incentives Apple to keep its original price. Samsung would then be confronted to either keep its price (1012CHF for the  Galaxy S9 )  or to lower it[1]. If Samsung is smart (and they are) they won’t respond by lowering their price. Sure, this might win them a larger profit that year, but the following Apple will follow their example leading to both lowering their prices (bottom right). This scenario destroys the phone market as less of the consumers money is going into purchasing a phone, irrelevant of the company. The profits of both companies in all scenarios are represented below.

Samsung keeps price Samsung lowers price
Apple keeps price Samsung Profits:          50$

Apple Profits:               50$

Samsung Profits:   70 - 30 = 40$

Apple Profits:                        30$

Apple lowers price Samsung Profits:          30$

Apple Profits:  70-30 = 40$

Samsung Profits :                  30$

Apple Profits:                        30$

In this game scenario there is no clear dominant strategy for each player. This is known as a mixed strategy. The optimal payoff of one company is entirely dependent on what the other does. Each company will do whatever it thinks the other will do. This will lead on the long run to two Nash equilibriums. Either both will keep their prices, or both will end up lowering them. Here it is clear both companies would rather stay in the first scenario where both keep their price and enjoy a larger profit than in if they entered into a price war[2].

Conclusion

The phrase “Sometimes you win sometimes you lose” hints a binary outcome to every game of play. Game theory suggest outcomes are dependent on payoffs which can be subject to change. The best decision is made when all players’ incentives are taken into play. As John Nash said, “The Best for the Group comes when everyone in the group does what's best for himself AND the group.” When life then throws you one of its game, stop to think “what are my payoffs and those of the other players? Is there anything I can do to change them?” Maybe then you will see how to change the payoffs, change the game and WIN!

Glossary

 Nash Equilibrium : where the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent's choice

Pareto Efficient Outcome : is an economic state where resources cannot be reallocated to make one individual better off without making at least one individual worse off. Pareto efficiency implies that resources are allocated in the most efficient manner, but does not imply equality or fairness

Dominant Strategies : occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play

Mixed Strategies : is any of the options he or she can choose in a setting where the outcome depends not only on their own actions but on the action of others.[1] A player's strategy will determine the action the player will take at any stage of the game.

Price Fixing : is an agreement between participants on the same side in a market to buy or sell a product, service, or commodity only at a fixed price, or maintain the market conditions such that the price is maintained at a given level by controlling supply and demand

References

Erickson, K. H. Game Theory a Simple Introduction. CreateSpace Independent Publ., 2013.

“Price Fixing.” Wikipedia, Wikimedia Foundation, 5 Nov. 2018, en.wikipedia.org/wiki/Price_fixing.

Staff, Investopedia. “Nash Equilibrium.” Investopedia, Investopedia, 19 Oct. 2018, www.investopedia.com/terms/n/nash-equilibrium.asp.

Staff, Investopedia. “Pareto Efficiency.” Investopedia, Investopedia, 27 June 2018, www.investopedia.com/terms/p/pareto-efficiency.asp.

“Strategic Dominance.” Wikipedia, Wikimedia Foundation, 31 May 2018, en.wikipedia.org/wiki/Strategic_dominance.

“Strategy (Game Theory).” Wikipedia, Wikimedia Foundation, 30 Oct. 2018, en.wikipedia.org/wiki/Strategy_(game_theory).

 

[1]For purpose of example the $187 price difference between the Iphone Xs and Galaxy S9 is insignificant. If you are already paying 1000 dollars for a phone perhaps a fifth more may not be a big difference.

[2]DISCLAIMER: This article does not suggest in any way that Apple or Samsung know what game theory is or that they follow any of the strategies proposed here.

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