Cryptoeconomic Theory: Game Theory Basics

Learn to Think Game Theoretically

This is part 4 of the weekly Cryptoeconomics series

This post will be split among three main sections:

  1. What is Game Theory?
  2. How it is used?
  3. Classifying Games.

1. What is Game Theory

When designing systems for organization, the goal is to structure social interactions in a way that leads to social order and good outcomes. To do so we need a framework for mentally approaching system design for humans.

Thankfully, for the last few decades, economists have been developing an arsenal of tools for this very purpose. We now have mathematical and theoretical approaches to predict how people will act, given certain adversarial conditions.

Game Theory is the science of logical decision making in humans, animals, and computers.
  • In public policy, game theory is used to predict how nations will act and react.
  • In war, game theory is used to predict the moves of the opponent.
  • In cryptography, game theory is used to predict potential cyberattacks.
  • In token design, game theory is used to predict the actions of token-holders in response to embedded incentives.
  • In financial markets, game theory is used to predict stock market decisions.

These predictions stem from the human tendency to maximize benefits and minimize losses, and thus can be calculated. Knowing how ‘players’ will act then leads to the resulting outcome of the ‘game’. This resulting outcome is named the Dominant Strategy, and we can use it to predict what decisions players will make.

2. How do we use it?

Here are some examples to help explain terminology:

In every game we have to specify, the players, their strategies and the payoffs. This is how to set up a basic Game Theory problem using an example of nuclear launch threat:

  1. Specify players, strategies, payoffs
  2. Create a normal form game

In academic literature, players are often abbreviated to i, strategy profiles are abbreviated to s, and payoffs abbreviated to u or π. A common nerdy way to describe what’s happening is:

Given any strategy profile s, each player i receives a payoff ui(s), which depend on the entire vector of strategies within that profile.

Also, payoff might mean several different things. It could mean the satisfaction or utility of an individual; it may stand for the monetary reward he/she receives; it could stand for the expected value of monetary reward or utility in an uncertain situation.

To predict the outcome of the interaction between the United States and Soviet Union above we need to find the Nash Equilibrium of the game.

Intro to Nash Equilibrium

The Nash Equilibrium is an outcome where each player has no incentive to change its behavior given the actions of other players.

The way we calculate Nash Equilibrium is as follows:

Reminder: the payoffs are structured with US being first, SU second. Like this — (United States payoff, Soviet Union payoff)
  1. Only look at the payoff of United States if they attack. We see -1 and 0. 0 is better, we underline 0
  2. Only look at the payoff of United States when they don’t attack. We see 4 and 2. 4 is better, we underline 4
  3. Only look at the payoff of Soviet Union when they attack. We see -1, and 0. 0 is better, we underline 0
  4. Only look at the payoff of Soviet Union when they don’t attack. We see 4, and 2. 4 is better, we underline 4
The game’s dominant strategies is to (Attack, Don’t attack) or (Don’t attack, Attack).

We’ll look at this example again later when drawing conclusions and making real-life predictions. What’s important is we maintain a standardized approach when evaluating game theoretic problems.

3. Classifying Games

When considering games:

  • Do the players have full or equal information?
  • Are the players’ interests in total conflict, or is there some commonality?
  • Are agreements to cooperate enforceable?
  • Are the moves in the game sequential or simultaneous?
  • Is the game played once or repeatedly, and with the same or changing opponents?

Do the players have full or equal information?

Imperfect vs Perfect Information

Imperfect information exists when the game has either external uncertainty (uncertainty about external circumstances like weather or the quality of a good) or strategic uncertainty (when each player is uncertain about the moves his opponent has made in the past or is making at the same time he makes his own move).

Perfect information exists if the game has neither external or strategic uncertainty.

Incomplete vs Complete Information

Incomplete information exist when one player knows more than another does (situations of asymmetric information). In such situations, the players’ attempt to infer, conceal, or sometimes convey their private information become an important part of the game.

Complete information exist when players have full & equal information.

Are the players’ interests in total conflict, or is there some commonality?

Basic Games

Remember the two basic game theory charts from last section? One is commonly known as pure conflict, and the other pure common interest

π symbolizes each players’ payoffs

Traffic jam is an example of common interest: jams are generally poor outcomes, and managing to avoid them would benefit everyone.

In other interactions, like settling on a price of a good to be exchanged or various zero-sum games, more for one means less for the other. These are examples of conflict.

Tragedy of Commons

Most games, however, have both elements of conflict, and common interest.

The Tragedy of Commons is a common problem where an individual’s Nash Equilibrium overwhelms the supply, and consuming an additional unit directly harms others.

This happens when certain resources are deemed free to players of the game. Outcomes occur such as:

  • Overfishing
  • Air pollution
  • No upkeep of public pastures
  • Traffic on public roads
  • Exponential population growth
  • Groundwater shortage in Los Angeles
  • Garbage in the ocean

We’ll use examples like these in a later article to show how a Nash Equilibrium directly contradicts the Pareto Optimal Outcome, and thus leads to coordination/cooperation failure.

Intro to Taxonomy of Games

The question of Are players’ interests in total conflict, or is there some commonality is the most basic tool to categorize games– It will help us begin distinguishing between

  • Invisible Hand games
  • Prisoner’s Dilemma
  • Pure Coordination games
  • Assurance Games
  • Battle of Sexes
  • Hawk-Dove
  • Chicken Games

Full overview of game theory taxonomy here soon.

Are agreements to cooperate enforceable?

Cooperative vs Non-Cooperative Games

Cooperative Games: Imagine an interaction for which it is the case that everything that both is affected by the actions of the players and is of concern to any of the players is subject to binding (meaning costlessly enforceable) agreement. This is termed a cooperative game. The term does not refer to the feelings of the parties about each other but simply to the institutional arrangements governing their interactions.

Non-cooperative Games: More commonly, however, something about the interaction is not subject to binding agreement. Such situations are modeled as noncooperative games.

Notice how this doesn’t necessarily refer to the ability or desire to cooperate, Cooperative Games simply refer to the enforcement level found in binding vs non-binding agreements.

In the social arrangement between an employer and employee, part of the interaction may be addressed cooperatively, as when an employer and an employee bargain over a wage and working hours. Other aspects of the same interaction may be noncooperative because of the impossibility of writing or enforcing the relevant contracts.

Examples include how hard the worker works or whether the employer will invest the resulting profits back into the company. When the employer is measuring productivity, he/she plays a cooperative game if he/she has a ticker at the door that tracks when the employee walks in and out. However, if the employer can’t verify the quality of work or effort put in, then the game is non-cooperative.

In technology jobs, and inside startups especially, it’s often difficult and costly to track worker productivity at work → making it a non-cooperative game. This is why we hear so often about ‘company culture’ and establishing a common ‘mission’. When an employer has no way enforce certain agreements, and rational employees can slack off without repercussions, therefore a narrative of ‘culture’ is used to imprint made-up social contracts.

Most commonly, social interactions are categorized based on whether the game representing it is cooperative or noncooperative and whether the payoffs of the game are common interest or conflict.

Are the moves in the game sequential or simultaneous?

There are two main games that differ based on ordering of decisions — Normal Form Games and Extensive Form Games.

Normal Form Games

Normal Form Game

This means that the time sequence of the actions taken by each player is not explicitly represented, the assumption being made that each player moves without knowing the move of the others.

The sequence of actions here is not important because we assume players are moving at the same time. The assumption is that one player would have full information on what the other player’s moves are.

Extensive Form Games

The games make explicit the order of moves, and which player knows what at each stage in the game. Moves made earlier in time need not be known by those making later moves.

Extensive Form Game

The extensive form conveys more information about the interaction in the sense that many extensive-form games may be represented by the same normal form game. When, as is common, the normal-form representation is used, this is because the additional information in the extensive form is thought to be irrelevant to how the game will be played.

Is the game played once or repeatedly?

A repeated game is an extensive form game that consists of a number of repetitions of some base game. Repeated games capture the idea that a player will have to take into account the impact of his or her current action on the future actions of other players.

Repeated games are one of the most powerful tools to preserve social order.

It’s why sexual relations with friends are often not worth it, why prisoners dilemmas can be overcome through system design, why going to war is very costly (but threatening war is very beneficial), and why monarchies sometimes work well.

There is also a further distinction between infinitely repeated games and finitely repeated games that we’ll discuss here soon.

Coming up : Prisoners’ Dilemma, Nash Equilibrium, and more deep dives.

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