Game Theory

Game theory explained using normal-form game

The best way to explain game theory is to use a normal-form game example. The normal form of a simple game is a four-square matrix that presents the personal payoffs for two players who are choosing between two decisions. Table 1 shows the concept of a payoff matrix, or normal form, for a simple game between two players. Notice that each player's outcome depends on their choice and the other player's choice.

Table 1. Concept of a normal form payoff matrix in game theory

Let's consider a scenario where both players choose A. Knowing that player 2 is choosing A, player 1 has two options. Either stick with A, in which case they both win, or choose to switch to B, in which case player 1 wins even more!

Now, this game happens to be symmetric. While player 1 realizes that switching to B can make them win even more, player 2 also thinks the same thing. So the rational outcome in this example is for both players to choose B. The result is that both players have a worse outcome than if both had remained at A.

A key factor in this particular game is that the players are not allowed to discuss their choices with each other in advance. That's why both players are in the dark about their opponent's choice. With this lack of information, it is not rational to choose A.

However, if the players could talk with each other, then any rational person would say "why don't they just agree to both choose A?" Well, check that knock at the door, it's the police, you are under arrest for collusion. Collusion, or price-fixing, is when firms conspire together to take advantage of monopoly power, rather than compete. When firms collude, the outcome is anti-competitive and consumers get hurt. Collusion is against the law in the U.S.

Game Theory concept and analysis

Game theory offers a way of modeling firms' decisions as optimal strategies in simple games. This allows economists to study market pressures and optimal strategies. Using this structure we can analyze the options players are considering and why they have the incentive to choose a particular option.

Table 2 shows a simple game. Notice that the payoffs are numbers. A higher number is a better payoff. If we think of each player as a firm, then these numbers might represent each firm's profit or loss. Each box with a set of numbers displays the outcome for Player 1 first, and then the outcome for Player 2.

Table 2. Example of a simple game

In this game, each player is presented with two choices. Naturally, a player will form a strategy to determine how they should play. Consider what player 1 would think about the game? Player 1 thinks to themself, "if player 2 chooses A, then I want to choose B, and if player 2 chooses B, then I still want to choose B." By doing this player 1 analyzes the optimal choices depending on how the other might play the game.

Behavioral Analysis and Dominant Strategy

In Table 2, we see that two players are each faced with two choices, and each player has an incentive to choose B in order to maximize personal profit, which ultimately causes them both to accept a fairly bad outcome. The outcome is nevertheless stable because each player can do no better considering the other player's choice.

Let's break down each step of the matrix to understand it better. The trick is to compare one player's options while holding the other's player's choice constant.

Table 3. Partial payoff matrix for player 1 assuming player 2 chooses A

Rationally, you decide that if player 2 has chosen A, you want to choose B. Now let's figure out what you should do if player 2 chooses B. If player 2 chooses B, then your choices and payoffs are given in Table 4.

In this scenario, you have no choice but to accept a loss. You can take a big loss by choosing A, or a loss that is slightly less bad by choosing B. The rational decision will be B.

Now player 1 has decided on their optimal strategy when taking player 2's choice as given. If player 2 chooses B, then play B. If player 2 chooses A, then play B. In fact, regardless of what player 2 does, play B. That choice always gives the better payoff between the two options.

When a player is better off choosing the same option in both cases, that is known as having a dominant strategy. If player 1 is to maximize their own personal gain, then they would always take B. Another way of thinking of it is that player 1 has no incentive to change.

What about player 2? Not every pair of opponents have the exact same payoffs every time. However, in this example, they do. Player 2's choices are an exact mirror of player 1's and will follow the same rational analysis. Therefore, player 2 makes the same decision and also has a dominant strategy of playing B.

An outcome of a game is a strategy for player 1 and a strategy for player 2. Both players choosing B is one possible outcome. It happens to be an equilibrium outcome. That's because even knowing for certain what the other player is choosing, both players are still happy with their choice. This is known as Nash Equilibrium, named after mathematician and Nobel Laurette John Nash.

In Table 2, the only Nash Equilibrium is where both players choose B and end up with -10. This is a pretty unfortunate outcome, but taking the other player's action as given, neither player is able to do any better.

When both players have a dominant strategy, then that outcome of the game is automatically a Nash equilibrium. However, a game can have multiple Nash equilibria. And a game can have one or more Nash equilibrium outcomes even if no one in the game has a dominant strategy.

How do economists know what choice players will make?

Economists always start with the assumption that individuals and firms are rational, utility- or profit-maximizing, and respond to incentives. The outcome of (-10,-10) in Table 2 is a result of rational self interest and imperfect information.

In a marketplace that rewards cooperation between firms, firms have a rational incentive to communicate with each other in order to get around this problem. This is called engaging in collusion, and in the U.S. there are legal repercussions for this kind of anti-competitive behavior. Having imperfect information about other firms is what keeps the marketplace competitive.

However, one of the main assumptions that economists make is that individuals are perfectly rational and utility-maximizing, and this may be a false assumption. It is often referred to as the imagined Economic Man or "homo economicus".

Example of Game Theory Approach

One of the most common non-market examples of game theory is the nuclear arms race that resulted in the aftermath of World War II. The Soviet Union had defeated the Axis forces in numerous Eastern European countries, while the Allied forces secured the Western European countries.

The two sides had rivaling ideologies and were hesitant to concede the land that they fought and died for. This led to a prolonged Cold War between the United States and the Soviet Union, where both countries tried to outcompete each other on military power to convince the other to back down.

In Table 5 below, we will analyze the payoffs both countries had using a 1-10 scale where 1 is the least preferred outcome and 10 is the most preferred outcome.

Table 5. Normal form payoff matrix in Cold War nuclear armament

It's important to note that the United States was more financially stable than the Soviet Union, predominantly because the Soviet Union had suffered in the war a lot longer, including invasions of its own land, and it had significant military and civilian casualties. This difference in financial stability can be seen in the asymmetric outcomes each country receives for the same actions. Disarmament provides a better outcome for both, as the money spent on weapons could be used elsewhere in a more productive economic market.

Now we can specifically examine the United States' decision by isolating Soviet Union's choice and respective payoffs, taking as a given choice that the Soviet Union makes.

Table 6. Partial payoff matrices for the United States

By isolating potential outcomes given a particular Soviet Union choice, the United States has a clear dominant strategy. In both cases, nuclear armament provides the United States with a better outcome than disarmament when holding the rival's decision constant. This can be seen numerically by comparing the numbers in Table 6 above.

Now we can specifically examine the Soviet Union's decision by isolating United States' choice and respective payoffs, taking as a given choice that the United States makes.

Table 7. Partial payoff matrices for the Soviet Union

In Table 7 above, while holding the United States' choices constant, we can see in both scenarios the Soviet Union has an incentive toward nuclear armament. Despite having slightly worse outcomes than the United States, it is still the better option to continue nuclear armament.

This resulted in a seemingly endless and globally destructive stalemate that significantly drained and reshaped both countries. The Soviet Union, while trying to maintain its military growth, was not able to also maintain its economy, which after enough time collapsed. The United States, in an effort to thwart the Soviet communist threat, engaged in multiple wars including the Korean and Vietnam war. These wars were extremely detrimental to the United States and offered little benefit aside from hurting the Soviets.

Looking back now it's easy to see that both countries would have been better off disarming and negotiating, so why didn't they? Well, they actually did negotiate several times, however, these negotiations only proved the pitfalls shown by game theory. When a disarmament negotiation occurred, that meant the payoff for reneging on the agreement was an outcome of 10!

Importance of Game Theory

Game theory has provided insight to economists in several classical settings not only in markets but also in international affairs. This section describes some of the important applications of game theory.

Game theory provides important insight into competitive interactions that occur within the marketplace. Firms in a crowded market have many factors to consider and the investments they make will always have varying returns. By modeling options using game theory, firms can determine the best strategies. Additionally, firms that can recognize when they are trapped in a losing situation can attempt to change the circumstances that led to the loss.

Firms that recognize this outcome through game theory can attempt strategies that mitigate the effects of competition, such as product differentiation. Firms can add features or establish quality through brand recognition to separate themselves from the competition. In the example above we see that firms' feasible choices are limited by competitive pressures, so firms attempt to alleviate competitive pressure by distinguishing their brand in a significant way. This leads to the concept of oligopolies.

Oligopolies

An oligopoly is a type of market that is dominated by a few very large firms, typically with differentiated products. It is a form of imperfect competition. These few very powerful companies can use their brand recognition to escape competition and therefore mitigate lose-lose scenarios. As we saw in the examples above, firms that are competing can struggle to find ways to invest that aren't dampened by competition. Using game theory to determine which business strategies yield the best results is part of what leads to the creation of oligopolies.

An example of an oligopoly, specifically a duopoly, is Coke and Pepsi in the market for caffeinated beverages. There are many other companies, but these two essentially monopolize the market. They are essentially competing only against each other. That's why this kind of market structure can be analyzed in a simple game with only two players. Analyzing the oligopoly setting with game theory has provided economists with a lot of insights about oligopolies.

Price Competition

A second common application is price competition. Firms have an incentive to undercut the competition by lowering their price. However, when all firms in the market respond in the same way, the result is very competitive prices. This means low profits for the firms, although it's a good outcome for consumers.

Advertising

Another common example is advertising. It is not clear that more advertising is beneficial for firms, but if a competing firm is advertising and you aren't, that is surely harmful. So we reach an equilibrium where so many firms are spending so much money on advertising even though it is costly and has dubious benefit.

International Affairs

Finally, during the Cold War between the U.S. and the Soviet Union, one world-destroying example from game theory provided valuable insight on the possible disastrous outcome from a global arms race among rational actors. The world consensus is that nuclear weapons should never be used, but each entity can achieve great strategic power from the appearance of military or nuclear strength as a deterrent. However, when rivaling entities both have nuclear missiles, neither can use them without mutual destruction, creating a stalemate. The irony is that both would prefer a non-nuclear stalemate, although private incentives lead both to deviate to the more expensive and deadly nuclear stalemate.

Types of Game Theory

There are many different types of games, whether cooperative or non-cooperative, simultaneous and sequential. A game can also be symmetric or asymmetric. The type of game that this explanation has focused on is a non-cooperative simultaneous game. That is a game where players are individually maximizing their self-interest and making choices at the same time as their competitors.

Sequential games are turn-based, where one player must wait for the other to make their choice. Sequential games can be applied to intermediary markets where firms elect to buy their raw materials from other firms, but they cannot take further action until the producer of the raw materials makes them available.

Cooperative game theory applies to why coalitions are formed in the marketplace, typically due to shared commodities or geographical proximity. An example of an international for-profit coalition is OPEC, which stands for Oil and Petroleum Exporting Countries. A cooperative game theory model can also be used to model the benefits of the North American Free Trade Agreement (NAFTA) between the US, Mexico, and Canada, or the creation of the European Union (EU).

The Prisoner's Dilemma

A very common game theory example is the Prisoner Dilemma. The prisoner's dilemma is based on a scenario in which two people are arrested for committing a crime together. The police have evidence to jail them both for a lesser crime, but in order to charge them on their most serious offense, the police need a confession. The police interrogate the criminals in separate rooms and offer them each the same deal: stonewall, and go to jail on the lesser crime, or testify against their co-conspirator, and get immunity.

The main conclusion from analysis of the prisoner's dilemma game is that each player's personal self-interest can lead to a collectively poor outcome for the criminals. In this game, both players have a dominant strategy to confess. Whether the co-conspirator confesses or not, it is always better to confess. In the end, both go to jail for the most serious offense, instead of remaining tight-lipped and getting a shorter prison sentence.

This analysis explains how two competitive firms that maximize their own individual profits can end up in an outcome that they may both be unhappy with. Of course, that's the benefit of competition. Both firms get fewer profits, but customers end up with lower prices.

Game theory gives a structure for economists to analyze competitive market behavior. Through the use of game theory, the most efficient outcomes can be more easily identified. Furthermore, games can show how certain decisions that lead to seemingly poor outcomes can arise from rational self-interest. Altogether, game theory is a useful tool in economics.

1. The Economic Man sourced from corporatefinanceinstitute.com