Introduction to Game Theory

By: Trishna Patnaik

Game theory is a theoretical framework for conceiving social situations among competing players. In some respects, game theory is the science of strategy, or at least the optimal decision-making of independent and competing actors in a strategic setting. Let’s have a look at a few examples.

The key pioneers of Game Theory were mathematician John von Neumann and economist Oskar Morgenstern in the 1940s. Mathematician John Nash is regarded by many as providing the first significant extension of the von Neumann and Morgenstern work!

The Basics of Game Theory

The focus of game theory is the game, which serves as a model of an interactive situation among rational players. The key to game theory is that one player’s payoff is contingent on the strategy implemented by the other player. The game identifies the players’ identities, preferences, and available strategies and how these strategies affect the outcome. Depending on the model, various other requirements or assumptions may be necessary.

Game theory has a wide range of applications, including psychology, evolutionary biology, war, politics, economics, and business. Despite its many advances, game theory is still considered a young and developing science. According to game theory, the actions and choices of all the participants affect the outcome of each!

Let’s Begin with the Nash Equilibrium

Nash Equilibrium is an outcome reached that, once achieved, means no player can increase payoff by changing decisions unilaterally. It can also be thought of as “no regrets,” in the sense that once a decision is made, the player will have no regrets concerning decisions considering the consequences involved.

The Nash Equilibrium is reached over time, in most cases. However, once the Nash Equilibrium is reached, it will not be deviated from. After we learn how to find the Nash Equilibrium, take a look at how a unilateral move would affect the situation. Does it make any sense? It shouldn’t, and that’s why the Nash Equilibrium is described as “no regrets.” Generally, it is considered that there accounts for more than one equilibrium in a game!

This primarily occurs in games with more complex elements than with two choices by two players. In simultaneous games that are repeated over time, one of these multiple equilibrium is reached after some trial and error. This very scenario of different choices overtime before reaching equilibrium is most often played out in the business world when two firms are determining prices for highly interchangeable products, such as airfare or even soft drinks.

Impact of Game Theory on Economics and Business

Game theory has brought about a revolution in economics by addressing crucial problems in prior mathematical economic models. For instance, neoclassical economics struggled to understand entrepreneurial anticipation and could not handle the imperfect competition. Game theory turned attention away from a steady-state equilibrium toward the very market process.

In business, game theory is definitely beneficial for modelling competing behaviours between economic agents. Businesses often have several strategic choices that affect their ability to realize economic gain. For example, businesses may face dilemmas such as whether to retire existing products or start developing new ones, lower prices relative to the competition, or go about employing new marketing strategies. Economists often tend to use game theory to understand oligopoly firm behaviour. It helps to predict likely outcomes when firms engage in certain behaviours, such as price-fixing and collusion.

Types of Game Theory

Although there are many types (e.g., symmetric/asymmetric, simultaneous/sequential, et al.) of game theories, cooperative and non-cooperative game theories are the most common. Cooperative game theory deals with how coalitions, or cooperative groups, interact when only the payoffs are known. It is a game between the coalitions of players rather than between individuals, and it questions how groups form and how they allocate the payoff among players.

Non-cooperative game theory deals with how rational economic agents deal with each other to achieve their own goals! The most common non-cooperative game happens to be the strategic game, in which only the available strategies and the outcomes that result from a combination of choices are listed down. A simplistic example of a real-world non-cooperative game is Rock-Paper-Scissors.

Examples of Game Theory: There are several “games” that game theory analyzes.

Below, we will just briefly describe a few of these.

The Prisoner’s Dilemma

The Prisoner’s Dilemma is the most well-known example of game theory. Consider the example of two criminals arrested for a crime. Prosecutors have no hard evidence to convict them. However, to gain a confession, officials remove the prisoners from their solitary cells and question each one of them in separate chambers. Neither prisoner has the means to communicate with each other. Officials present four deals, often displayed as a 2 x 2 box.

If both confess, they will each receive a five-year prison sentence.

If Prisoner 1 confesses, but Prisoner 2 does not, Prisoner 1 will get three years and Prisoner 2 will get nine years.

If Prisoner 2 confesses, but Prisoner 1 does not, Prisoner 1 will get 10 years, and Prisoner 2 will get two years.

If neither confesses, each will serve two years in prison.

The most favourable strategy is then not to confess. However, neither is aware of the other’s strategy and without certainty that one will not confess, both will likely confess and receive a five-year prison sentence! The Nash equilibrium suggests that in a prisoner’s dilemma, both players will make the move that is best for them individually but worse for them collectively.

The expression that is “tit for tat” has been determined to be the optimal strategy for optimizing a prisoner’s dilemma. Tit for tat was introduced by Anatol Rapoport, who developed a strategy in which each participant in an iterated prisoner’s dilemma follows a course of action consistent with his opponent’s previous turn. For example, if provoked, a player subsequently responds with retaliation; if unprovoked, the player does cooperate.

Dictator Game

This is a simple game in which Player A must decide how to split a cash prize with Player B, who has no input into the Player A’s decision. While this is not a game theory strategy per se, it does provide some interesting insights into people’s behaviour. Experiments reveal about 50% keep all the money to them, 5% split it equally, and the other 45% give the other participant a smaller share.

The dictator game is very closely related to the ultimatum game, in which Player A is given a set amount of money, part of which has to be given to Player B, who can accept or reject the amount given. The catch here is if the second player rejects the amount offered, both A and B get nothing. The dictator and ultimatum games hold important lessons for issues pertaining to causes such as charitable giving and philanthropy.

Volunteer’s Dilemma

In a volunteer’s dilemma, someone has to undertake a chore or a job for the common good. The worst possible outcome is realized if nobody volunteers. For example, consider a company in which accounting fraud is rampant, though top management is unaware of it. Some junior employees in the accounting department are aware of the fraud but hesitate to tell top management because it would result in the employees involved in the fraud being fired and most likely to be prosecuted!

Being labelled as a whistleblower may also have some repercussions and consequences down the line. But if nobody volunteers, the large-scale fraud may result in the company’s eventual bankruptcy and the loss of everyone’s jobs.

The Centipede Game

The centipede game is an extensive-form game in game theory in which two players alternately get a chance to take the larger share of a slowly increasing money stash. It is arranged so that if a player passes the stash to his opponent who then takes the stash, the player receives a smaller amount compared to if he had taken the pot.

The centipede game concludes as soon as a player takes the stash, with that player getting the larger portion and the other player getting the smaller portion!  The game has a pre-defined total number of rounds, which are known to each player well in advance.

Limitations of Game Theory

The biggest issue with game theory is that, like most other economic models, it relies on the assumption that people are rational actors that are self-interested and utility-maximizing. We are social beings who do cooperate and care about the welfare of others, often at our own expense! Game theory cannot account for that fact that in some situations we may fall into Nash equilibrium, and other times we may not, depending on the social context and who the players are.

Repeated Oligopoly Games

The prisoners’ dilemma was played once, by two players. The players were given a payoff matrix; each could make one choice, and the game ended after the first round of choices.

The real world of oligopoly has as many players as there are firms in the core industry. They play round after round: a firm raises its price; another firm introduces a new product, when the first firm cuts its price, then a third firm introduces a new marketing strategy, and so on. An oligopoly game is a bit like a baseball game with an unlimited number of innings one firm may come out ahead after one round, then another will emerge on top another day! For example: In the computer industry game, the introduction of personal computers changed the rules. IBM, which had won the mainframe game quite handily, struggles to keep up in a world in which rivals continue to slash prices and improve quality.

Oligopoly games may have more than two players, so the games are far more complex, but this does not change their basic structure. The fact that the very games are repeated introduces new strategic considerations. A player must consider not the ways in which its choices will affect its rivals, but how its choices will affect them in the future as well!

Let us keep the game simple now, and consider a duopoly game. The two firms have colluded, either tacitly or overtly, to create a monopoly solution. As long as each player upholds the very agreement, the two firms will earn the maximum economic profit possible in the very enterprise.

Decision Making Process

Game theory is concerned with decision-making in an interactive world such that the best decision of every decision-maker depends on what decisions others make! As a result, everyone in this interactive world, for advancing one’s own self interest, will need to predict decisions of others.

Game theory officially entered the world in the year 1944 with the publication of the magnum opus in game theory, “Theory of Games and Economic Behavior”. This was a joint collaboration between an Austrian economist Oskar Morgenstern, and John von Neumann – a universally acclaimed genius, polymath and polyglot from Hungary.

Von Neumann was an undisputed genius, but he was a mediocre poker player and quickly realised that the probability theory cannot help one win poker games! His great appreciation for sketchy information, second-guessing and unpredictability of poker games laid the very foundation of game theory: how poker players can hide information by strategically releasing information through their moves and prompting mistakes from rivals.

In other words, he formalised how poker players can “bluff” their rivals by playing on a string of strategies that is supposed to deceive their rivals and hide information and finally win the game.

The legacy of John Nash in game theory is a unique and attainable Nash equilibrium, and game theory thus became totally clinical and totally removed from the real world. The only exceptions among accomplished economists today are two other Nobel-laureates Thomas C. Schelling and Roger Myerson.

If We were all Better People the World would be a Better Place

Some of the power and meaning of game theory can be illustrated by assessing the statement “If we were all better people the world would be a better place.” This may seem to you to be self-evidently true. Or you may even recognize that as a matter of logic this involves the fallacy of composition: just because a statement applies to an individual person it need not apply to the entire group.

Game theory can give precise meaning to the statement of both what it means to be a better people and what it means for the world to be a better place, and so makes it possible to prove or disprove the same statement. In fact the statement is false, and this can be shown by a variation of the Prisoner’s Dilemma.

What do We Conclude?

The key to game theory and towards understanding why better people may make the world a worse place is to understand the delicate balance of equilibrium. It is true that if we simply become more caring and nothing else happens the world will at least be no worse. However: the paradox initiates that if we become more caring we will wish to change how we particularly behave!  So when we both try to do this at the same time, the end result may make us all worse off. So that is what exemplifies and contradicts the Game Theory at the same time!

About the Author

Trishna Patnaik, a BSc (in Life Sciences) and MBA (in Marketing) by qualification but an artist by choice. A self-taught artist based in Mumbai, Trishna has been practising art for over 14 years. After she had a professional stint in various reputed corporates, she realised that she wanted to do something more meaningful. She found her true calling in her passion that is painting. Trishna is now a full-time professional painter pursuing her passion to create and explore to the fullest. She says, “It’s a road less travelled but a journey that I look forward to everyday.” Trishna also conducts painting workshops across Mumbai and other metropolitan cities of India.

Trishna is an art therapist and healer. She works with clients on a one on one basis in Mumbai. Trishna fancies the art of creative writing and is dappling her hands in that too, to soak in the experience and have an engagement with readers, wanderers and thinkers.

Featured image via Pixabay.