The Basics of Game Theory And Associated Games

The Basics of Game Theory And Associated Games

By Dr. Nicholas Milovsky.

Game theory is a fascinating field that has been around since the 1920s and has applications in many areas and can be seen in action every day. In essence game theory is about making decisions when the fortunes of participants are interlinked. It can be applied to a range of diverse fields including economics, politics, diplomacy, law, psychology and biology. When the 2001 movie A Beautiful Mind was made, starring Russell Crowe as the economist and game theorist, John Nash, it introduced game theory to laypeople. However, many of us have used game theory in our every day lives without necessarily knowing it.

First developed by a mathematician, John von Neumann and an economist, Oskar Morgenstern, game theory came to prominence with their landmark book on the subject Theory of Games and Economic Behaviour, published in 1944. Since then a great deal of research has been done and eight game-theorists have won the Nobel Memorial Prize in Economic Sciences.

One of the cornerstones of game theory is the zero sum game (1,-1), where the gains on one side are exactly equal to the losses on another; when one side increases or decreases by a certain amount it has the opposite effect on the other. Consequently, when both positive and negative are added it equals to a sum of zero. Game theory has value in many situations where decisions are being made in situations of uncertainty, with poker being a classic example.

Cooperative and noncooperative games Fundamentally, there are two wings of game theory - cooperative and noncooperative. A cooperative game is one where two parties work together to achieve a best outcome, while a noncooperative is one where the parties do not.

Cooperative games focus on the coalition result while noncooperative focus on the individual players’ payoffs and trying to predict what strategies players will choose. They are usually appropriate to situations where the procedures and rules are clearly defined and bargaining can take place, such as elections, auctions and some types of Internet sales. Similarly, they are suited to situations like markets where the details of transactions are not clear, such as who is making offers or when transactions will be completed. A non-cooperative game is one in which there is no form of negotiation or binding contracts in the game. It is not that players do not cooperate, but rather any cooperation must be self-enforcing.

Cooperative Games The underlying question for cooperative games is what payoff will players get for different coalitions of actions. Players can coordinate their strategies and share the payoff. A real life cooperative game scenario would be where a supplier and a customer negotiate an agreement on price. As explained by John C.S. Lui of the The Chinese University of Hong Kong, by extension these coalitions “can make binding agreements about joint strategies, pool their individuals agreements and redistribute the total in a specified way.�

A famous example of this type of game is the stag and rabbit hunt. In the scenario a stag provides more meat (4 points) compared to a rabbit (1 point). But hunting a stag requires cooperation, so that if Player 1 hunts the stag while Player 2 hunts the rabbit, then Player 1 will fail (0 points). However, if they both hunt the stag they will each get 2 points (2, 2). If Player 1 trusts that the other will hunt the stag then s/he should hunt the stag because s/he will get 2 points, whereas if s/he thinks Player 2 will hunt the rabbit then s/he should also hunt the rabbit and get 1 point (rather than hunting the stag, failing and getting zero points).

Player 2 hunts the stag

Player 2 hunts the rabbit

Player 1 hunts the stag

4, 4

0, 1

Player 1 hunts the rabbit

1, 0

1, 1

Another popular example of a cooperative game is chicken. This is where two players are rushing towards each other in cars with the options to swerve or hold their course. The best outcome for Player 1 is to hold their course while their opponent swerves. The next best outcome is if both players back down but the worst is if neither backs down. Essentially, it’s a high risk situation.

Swerve

Hold course

Swerve

0, 0

-1, 1

Hold course

1, -1

-10, -10

Cooperative game theory and particularly the game of chicken can be applied to the situation in North Korea, where brinkmanship has been utilised as a negotiating strategy. In order for this to work however, the threat wielded by one party needs to be credible. In the event that a party cannot bring force to bear they will likely lose the battle. The latter might apply to a union negotiation where an employer threatens a union with strike busting legislation but which in reality is an empty threat.

Noncooperative Games Noncooperative games deal with how rational individuals interact with each other in order to achieve their goals. A typical scenario would be the marketplace where a firm sets prices according to what it thinks its competitors will do but does not enter into an agreement with them.

A famous example of noncooperative game theory is one called the Prisoner’s Dilemma. In this scenario two prisoners who are partners have been arrested by the police for a crime. They are placed in separate cells and offered an opportunity to confess. If neither confesses then they will both go free and will split their booty equally. However, if one confesses and the other does not then the person confessing will be set free and will get all of their swag. The other prisoner will then go to prison. The final option is for both prisoners to confess, wherein they will both go to prison but for a reduced term.

The variations of the choices of the Prisoner’s Dilemma can be shown as follows:

P 2 - Does not confess

P 2 - Confesses

P 1 - Does not confess

3, 3

0, 5

P 2 - Confesses

5, 0

1, 1

P 1 = Prisoner 1 P 2 = Prisoner 2

Here we ascribe a number to the value of each action. Therefore, if both prisoners go free they end up with an equally satisfactory outcome (3, 3). By confessing, Prisoner 1 will go free if Prisoner 2 doesn’t confess (and will get a value of 5). However, if Prisoner 2 also confesses they will both go to prison and Prisoner 1 will end up with a value of 1. The safest option for Prisoner 1 is to confess because then s/he will get 5 at best and 1 at worst (thus avoiding a zero result). However, if both acted for the common good and neither confessed they would be sure to end up with a middling good result (3 each).

A development of the Prisoner’s Dilemma game is entitled the Pride Game. This adds an extra dimension where a proud individual is one who refuses to confess unless it is in retaliation against a prison who confesses. In this game the best outcome is achieved for Prisoner 1 if

they are proud, facing a prisoner who is not proud and Prisoner 1 does not confess.This type of situation, where players try to win based on the knowledge they hold, including what they have learned about their opponent, is called a Nash Equilibrium. Basically it describes a situation where a player cannot improve their situation by unilaterally changing their strategy.

Further development of the theory, known as a mixed Nash Equilibrium, allowed a player to change strategy based on assigning probabilities to players’ strategies. The Nash Equilibrium is useful in situations where there are several decision makers, such as games like poker. It tries to predict what will happen if several participants make decisions at the same time, taking into account other people’s actions.

David K. Levine of the Department of Economics, UCLA, has said that the key to game theory is understanding the importance of the balance of equilibrium and how this can be affected by the decisions we make. Your actions may change the way I decide to act, which will in turn affect the eventual outcome.

Applications of Game Theory Game theory can be seen in real world situations almost everywhere you look. Making decisions about how other people will act ensures that the majority of people do not volunteer for charity. The rationale for why is often a combination of believing that other people are basically good and will help by volunteering (basing our decisions on information or perceptions of other people or participants). In a similar way, it explains why some countries do not invest in their military defence structure, believing that the U.S. will come to their aid to protect their own interests.

Pure Strategy A pure strategy approach means deciding in advance how a game or a situation will be played.

Here the individual(s) know what moves will be made for any scenario. It prevents random moves from being made and does not take into account moves made by other players. A pure strategy is sometimes used by poker players and has been called a game theory optimal strategy (GTO). Although this makes it sound quite technical it basically means deciding in advance how to play depending on the cards that are shown. So, if a player was dealt a high value pair they might always raise, whereas with a connected suit (e.g. 7,8 clubs), they might call.

A pure strategy bases card decisions on what will make a profit, regardless of an opponent ‘s actions. However, profit does not necessarily mean making money in a game - it could also mean losing as little as possible. In a poker game if a player was a big or little blind using a GTO would mean cutting their losses and folding on a losing hand. A GTO strategy is defensive in nature - a player is playing not to lose. Therefore the result is often a case of winning small amounts but also not losing very much. Some players believe that a GTO strategy is ideal in games against better players as it takes the opponent factor out of the game. One of its most famous proponents is Chris ‘Jesus’ Ferguson who won the World Series of Poker in 2000 using a GTO strategy. According to a 2009 New Yorker article on Ferguson the GTO strategy protects a player because it prevents him or her from falling into the trap of making impulsive judgment calls that ignore relevant information. However, while a strategy based on GTO will negate bluffing by another player it may also disregard important information and may lead the player down the path of predictability.

Mixed Strategy Why do people bluff in sports or use plays that are weaker than the best they could choose?

Basically, they are applying a mixed game theory strategy - changing their tactics as the situation develops. A penalty kicker might shoot to the goalkeeper’s favoured right hand, while in a long volley a tennis player might play a surprise shot to their opponents forehand. One famous example of a mixed strategy was in the 1989 French Open when the then 17-year old Michael Chang beat Ivan Lendl, the world No.1, utilsing a number of unorthodox shots, including ‘moon shots’ and most famously serving underarm in the final set. In poker most players heavily rely on a mixed game, bluffing at times and holding strong hands at others.

If a player followed the same strategy at all times their opponent would anticipate their future moves and would then counteract them successfully. However, making an atypical move serves to ‘mix it up a little’ and ideally deceive the other player(s). The reason a mixed strategy is often seen in sports is because it usually works best when the participants’ objectives are strictly opposed, which happens most often in sports.

According to David Sklansky, a professional poker player and winner of three World Series of Poker bracelets, winning at poker is influenced by how much an individual can change the way their opponent plays while refusing to allow themselves to be put off their game in return. This kind of thinking is the inspiration for bluffing, slow playing and other forms of deception.

Some poker players employ an exploitative strategy, which is the opposite of a game theory optimal strategy and focuses on exploiting opponents’ weaknesses. It often plays out that an exploitative strategy will entail calling all or most of the time (80 - 100 percent). Quite aggressive as an approach, it can be risky because it relies on the assumption that opponents will be cowed into folding in the face of their opponent’s hardball style or alternatively, have a weak hand and are bluffing. The danger for a player using an exploitative strategy is that once other players realise that this is what a player is doing it quickly falls apart as they call or raise, when they have a good hand.

Information and incentives This aspect ties in two of the key elements of game theory - information and incentives. One player has more information than the other, while the other is in a position to give incentives for cooperation. Both parties also need each other and so must find a way to cooperate with one other. An example, as outlined in a PBS article, is where a software developer creates an app that has fantastic profit potential. The investors that the developer approaches have the capital but do not know anything about the product so they cannot judge its value. Instead they insist that the developer invests some of his own money in the venture. In this situation one player has the information, while the other employs the incentives. The developer knows the true value of their idea, while the investors have the money. Sometimes known as information economics, this application of game theory has been used in many fields, including corporate governance, industrial organisation, politics and law. Three of its originators, George Akerlof, Michael Spence and Joseph Stiglitz won the Nobel Memorial Prize in 2001.

How game theory might save you money The next time you are buying a car, try following these tips posted on Business Insider. Rather than going to a dealer and engaging in a negotiating dance with the salesman, instead identify each dealership in a defined area that sells your car of choice. Then call each one and explain that you will buy from whichever dealer gives you the lowest price. The most likely scenario is that the dealer will try to undercut his competitors. Bu because he is dealing with imperfect information and does not know what they are charging s/he will more than likely offer a lower price than s/he normally would in that situation. It is a strategy that works!

Sources

www.brown.edu www.businessinsider.com www.cse.cuhk.edu.hk www.fulltiltpoker.com www.gametheory.net www.gametheorystrategies.com/ http://increaseyourpokeriq.com http://levine.sscnet.ucla.edu micro.blogspot.com/2008/11/cooperative-versus-noncooperative-game.html www.newyorker.com www.pbs.org www.theguardian.com http://stonecoldbluff.co.uk www.uib.es www.wikipedia.org