Introduction to Game Theory in Finance

Game Theory is a method of modeling the interaction between two or more players in a situation with particular rules and expected outcomes.

It is helpful in many fields, but mainly as a tool in economics. Game Theory helps with the fundamental analysis of industries and the interactions between two or more companies.

Theoretically, games can have an infinite number of players, but we usually look at them in the context of two players. A simple example is a two-player game with sequential turns.

What Is Game Theory?

Game Theory is a theoretical framework to simplify social interactions between two or more competing players. It is a set of simple methods that help us solve confusing real-life situations, from Mergers & Acquisitions (M&A) to new product releases and others.

Mathematician John von Neumann and economist Oskar Morgenstern pioneered the theory in the 1940s. Their work followed the premise that we can lay out simplified scenarios and predict the outcomes.

The theory looks at the science of making optimal strategic decisions in various settings.

The critical point is that the payoff for one player is ultimately contingent on the other players’ strategy. The actions and choices of each participant affect the outcome for all of them.

We can use Game Theory to help figure out the most likely outcomes whenever we have a situation with quantifiable results for two or more strategic decision-makers or players in a game context.

Impact on Economics and Finance

Game Theory revolutionized economics and business analysis by addressing critical issues in the popular mathematical models. For example, neoclassical economists struggle to account for the concept of imperfect competition fully. Game Theory improves on that by switching the focus from constant equilibrium to analyzing the actual market process.

In business, we use it to model the competition between companies. Businesses usually have a set of choices in terms of strategy. These choices impact the firm’s ability to generate profits. Examples can be developing new products, lowering prices to gain a competitive advantage, working on new distribution channels, and others.

Nash Equilibrium

An essential concept within Game Theory is the Nash Equilibrium, which represents a stable state in a game, also known as a ‘no regrets’ state.

It is named after John Nash, who got the Nobel Prize for it in 1994. The concept represents an outcome within the game, at which point no player can increase payoff by changing their strategic decisions. Once we reach such a state, we usually don’t deviate from it. Unilateral moves no longer affect the situation, so it makes no sense to make them. And this is why we consider it a ‘no regrets’ state.

A set of strategies is at a Nash Equilibrium if each is the best response to the others. If all players operate on a Nash Equilibrium strategy, they have no incentive to deviate, as discussed above. Each player has adopted a plan that’s the best course of action based on what the others are doing.

Usually, a game can have multiple equilibriums. The Nash Equilibrium is the crucial solution concept in a non-cooperative (adversarial) game.

Assumptions and Limitations

As with any other economic theory, we need to be aware of the underlying assumptions that support Game Theory.

• All players are utility-maximizing rational economic agents;
• Players have the complete set of information about the game, its rules, and their consequences;
• In the general case, there’s no communication between participants;
• We know the possible outcomes in advance and cannot change them during the course of the game;
• In theory, the number of players can be infinite, but we analyze most models in the context of two players.

Participants in the game act rationally and aim to maximize their profit.

We assume the listed payoffs are the total benefit associated with the particular outcome.

The theory’s primary limitation lies with the assumption that people are rational and self-interested and act to maximize their payout. Game Theory cannot fully account that people are social beings, and sometimes we wish others’ welfare, sometimes even at our own expense.

Game Theory Classification

We can classify games based on a few aspects.

Cooperative and Non-cooperative

A cooperative game is one between groups of players. Such an analysis looks at how individuals form groups and how they allocate payoffs between themselves. In the premises of a cooperative game, we only know the payoffs.

A non-cooperative game, on the other side, looks into how rational individual agents interact with each other to achieve their own goals. The most common is the strategic game, where we know the available strategies and the possible outcomes from choice combinations. An example of that is Rock-Paper-Scissors.

Symmetric and Asymmetric

A symmetric game is a setting where playouts mainly depend on the player’s strategy, not on the others. On the other hand, in asymmetric games, the payouts vary among the participants. Even with the same strategy, two players will get different outcomes.

Zero-sum and Non-zero-sum

In a zero-sum ruleset, one player’s gains balance out with the other’s loss and vice versa.

A non-zero-sum game has no restriction in terms of potential gains. One player’s payout does not require the other player to lose, meaning we can have a win-win scenario.

Simultaneous and Sequential

A game where all players make their decision simultaneously, or without prior knowledge of other players’ decisions, is a simultaneous game. On the other hand, a sequential game is one where players take turns to decide or know other players’ choices.

Perfect and Imperfect Information

In a game with perfect information, all players have access to the same set of data. In a setting with imperfect information, each participant has access to proprietary information.

The Prisoner’s Dilemma

This is the most famous example of the Game Theory application. We often use it to illustrate the strategic choices in the premises of oligopolies.

We have two prisoners (Walter and Jessy) that have been caught for a petty crime, facing a 4-year sentence. The police detective suspects them of a more severe crime, which would increase their sentence to 15 years.

He separates the two prisoners so that they can’t communicate and then offers them a deal. This deal illustrates the possible outcomes:

• If both confess to the more severe crime, they will get a reduced sentence of 6 years each
• If both deny, they will get four years on the petty crime, but most probably won’t be sentenced on the more serious crime
• If one confesses and the other denies, they will get two years and 15 years, respectively.

Based on this information, we can go ahead and build our payoff matrix:

We can immediately notice that the payoff for Walter and Jessy depends on what the other one does.

To avoid the potential 15 years, the safest bet appears to be to confess and get six years.

If Walter and Jessy find a way to communicate, they can agree to deny and only get four years for the petty crime. However, each one has an incentive to betray the agreement and get only two years by confessing. Fearing this betrayal, the safest option remains to confess.

There are two strategies we can apply here.

Maximax

With this approach, we attempt to get the maximum possible benefit, even if this also means that a highly unfavorable outcome is possible. Analysts often refer to it as ‘best of the best.’ Some consider it overly optimistic and even naÃ¯ve.

The best outcome of confessing is two years (if the other one denies). The best outcome of denying is six years (if both Waler and Jessy deny). The ‘best of the best’ is to confess.

Maximin

This approach looks at the ‘best of the worst.’ It’s a common strategy when we do not trust the other players to honor their word or agreements.

The worst outcome of confessing is six years (if both Walter and Jessy confess). The worst result of denying is ten years (if the other one confesses). The ‘best of the worst’ is to confess.

The dominant strategy is the one yielding the best outcome regardless of the choices of the other players. In this case, it’s to confess, and both approaches lead to the same decision. In the Prisoner’s Dilemma, confessing is the Nash Equilibrium, as it’s the best outcome, considering the other player’s possible choices.

We can take the above example and apply it to any similar situation.

ACME Ltd. is looking at its pricing strategy for a specific product. They can raise the price or lower it, and so does their competitor ABC Ltd.

We know these are the two leading players in the market. If both raise the prices, their revenue from the said product will be \$ 10m each. If they lower them, each company will get \$ 6m in revenue.

The management of ACME Ltd. Has colluded with ABC Ltd.’s management, made an informal agreement to raise the price, and generated more revenue.

However, both managers know that if they break their promise and lower the price, while the other company raises it, they will get a competitive advantage, reaching \$ 19m in revenue. The other company (that raised the prices) will then realize only \$ 3m in revenue.

Here’s our payoff matrix for this scenario:

Let’s apply the same two strategies and identify the best course of action.

Maximax

The maximum payoff if we raise the prices is \$ 10m. If we lower the prices, the maximum outcome is \$ 19m (when the competitor raises the prices). Following the ‘best of the best’ logic, we will lower the prices.

Maximin

The worst outcome of raising the prices is \$ 3m (when the competitor lowers them). The worst result of reducing the prices is \$ 6m.

The dominant strategy is to lower the prices, supported by both the maximax and the maximin approaches.

Conclusion

The primary use of Game Theory is to describe and model the behavior of human populations. It is a widely-popular method within mathematical economics.

We use it to attempt to explain and understand the decision-making process and strategy between rational individuals. The goal is to identify the strategic choices a player should make to maximize their chance to succeed both logically and mathematically.

In finance, we can use it in many settings like:

• Investment banking;
• Mergers & Acquisitions;
• Asset pricing;
• Portfolio management.

The theory helps us to identify the optimal rational strategy in various scenarios.

FCCA, FMVA

Hi! I am a finance professional with 10+ years of experience in audit, controlling, reporting, financial analysis and modeling. I am excited to delve deep into specifics of various industries, where I can identify the best solutions for clients I work with.

In my spare time, I am into skiing, hiking and running. I am also active on Instagram and YouTube, where I try different ways to express my creative side.

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