Nash equilibrium and blockchain platforms: a token curated registry example
Nash equilibrium is an incredibly useful concept for individuals working in blockchain. While many of those working in the space recognize the importance of game theory, few are demonstrating — in their whitepapers or elsewhere — that the behaviors their platforms are created to induce are indeed equilibria on those platforms. In this post, I’ll discuss the basics of what constitutes a Nash equilibrium and how to verify that the desired outcomes on your platform are in fact equilibria.
What is the precise definition of a Nash equilibrium and when should this concept be applied?
A Nash equilibrium exists in a strategic situation that economists call a game.
What is a game?
A game satisfies these criteria:*
- A game must involve two or more active participants.
- Each participant is going to choose among several actions available to her.
- The goal of each participant is to choose the action or mix of actions that maximize her payoffs.
- Each participant’s choice of action determines the payoff that she gets in the game, and the payoffs of the other participants in the game.
- All participants choose their actions in isolation, and their choices are only revealed to each other after all choices are made.
Point 4 is critical. When a participant is playing a game, her payoff-maximizing choice depends on what she believes (or knows) the other participants will do, and she knows her choice will affect the optimal choices of her opponents.
The payoffs in a game can be anything that can be represented by numbers — revenue, profit, tokens, chocolate bars, or anything else that the participants would prefer to have more of. One of the reasons game theory has become so immensely popular is that games can be used to represent so many strategic situations.
A common mistake that organizations make when trying to apply game theoretic modeling is that they assume payoffs = tokens. In order to accurately represent a strategic situation, payoffs must capture (to the extent possible) all the gains that participants get from each action combination — including non-monetary payoffs (such as reputation) and payoffs from outside the platform.
Games are typically represented in the following matrix form. Let’s consider the game below.
In this game, there are two players: Player 1 and Player 2. Player 1 is going to choose action A or action B, and Player 2 is going to choose action C or action D. This matrix shows the gains or losses to each player depending on which actions both players choose. For example, if Player 1 selects action A and Player 2 selects action C, then both players get a payoff of 2. If Player 1 selects action B and Player 2 selects action C, then player 1 gets a payoff of 1 and player 2 gets a payoff of 0.
When does a Nash equilibrium occur?
Given a situation that can be represented by a game like this, we naturally want to be able to predict which actions each participant will choose, and therefore which outcome we should expect to see. Nash equilibrium is one way to predict this outcome.
The Nash equilibrium of a game occurs when each participant cannot increase her gains in the game by choosing a different action (or mix of actions) than she is choosing, given the choice of actions of the other players in the game.
In other words, given everyone else’s choices in the game, I have no reason to change my own. Therefore, the game is in equilibrium.
Let’s revisit our game from before.
There are two Nash equilibria here.
- Player 1 selects action A and Player 2 selects action C.
- Player 1 selects action B and Player 2 selects action D.
In both cases, neither player wants to change his or her choice of action given the choice of the other. Consider the first equilibrium. Currently Player 1 is getting a payoff of 2 from choosing action A. Her only option is to switch to action B, in which case she would get a payoff of 1. This is clearly worse than a payoff of 2, so she will stick with action A. Similarly, Player 2’s only option is to switch from action C to action D, which would decrease her payoff from 2 to 0. This is also worse for her, so she will stick with strategy D. The same reasoning applies to the second Nash equilibrium.
In the game above, each Nash equilibrium consists of every player choosing a single action. In more complicated games, each player’s optimal behavior can actually be to randomize over actions. This is called a mixed-strategy Nash equilibrium. It is possible to prove mathematically that every game has at least one (pure- or mixed-strategy) Nash equilibrium.
Why does Nash equilibrium matter for blockchain?
Strategic games are everywhere in blockchain ecosystems — in the design of markets, in consensus algorithms, and in the decisions that users and miners make of whether to act in the best interest of the platform community. Users and miners participating in blockchain ecosystems are not controlled by a centralized authority: they act in the way that maximizes their own outcomes, taking into consideration what they expect the other participants to do, exactly as game theory specifies.
Developers create a game when they decide what actions are available to participants and how these actions will be rewarded in tokens or other outcomes. They need to design their platforms so that the outcomes they want to occur are equilibria of the strategic games that their participants are playing. This is much more complicated than it sounds, and design choices can easily induce unintended consequences. It is important to check whether the desired behavior is, in fact, an equilibrium of the game you have created, and whether there are equilibria of that game that are undesirable.
An example of this type of game design is token curated registries, which aim to use game theory to decentralized the curation of ranked lists. Rather than relying on a list owner (such as a newspaper editorial team) to rank items by quality, users of the registry vote on which content should be accepted to the list. The token economics are supposed to guarantee that users only accept high-quality content to the list and don’t accept low-quality content. The claim is that since users must buy tokens to submit content to the list and stake tokens to vote on content, and the token will increase in value as the list becomes more popular, users will want to maximize the popularity of the list, and they will do this by voting for the highest-quality content.
Obviously, it is not this easy — there are a lot of hidden assumptions in here that need to be verified. This offers more general lessons for blockchain organizations.
Lesson 1: It is not enough to state that an outcome is an equilibrium — you must show that that is the case. Games similar to token curated registry structures, in which curators are rewarded when interest in their content grows, exist outside of blockchain — cable or online news platforms, and Reddit are all examples. In some of these platforms, curators do not choose the content they believe to be high quality, but instead choose content that the audience will like.
Lesson 2: Before determining the equilibrium, you need to outline what the possible actions are for each participant, and their payoffs from choosing each. A token curated registry aiming to include only the best restaurants in New York, for example, may have a small but devoted readership who enjoy a list of fine dining restaurants. However, the list might attract a much larger audience and more interest in its token if the list is filled with cat memes and Times Square chain restaurants. The developer of the registry needs to understand how the size and composition of the audience affects the payoffs of the token holders.
Lesson 3: After outlining the possible actions of the participants and their payoffs from each, the developer must verify that the desired outcome is an equilibrium of the game. Consider again the token curated registry of high-quality New York restaurants. The developer needs to verify that that, as long as the majority of token owners are voting to include ABC Cocina and to reject Guy’s American Kitchen, none of the voters in this majority would get a higher payoff from changing their votes and including the low-quality content instead.
Lesson 4: Many games have multiple equilibria — you need to verify that the equilibrium you want to occur will be selected. For example, a token curated registry could have two equilibria: the “good” equilibrium, where token-holders want high-quality content, vote to include it, and have attracted a customer base that values this content, and a “bad” equilibrium, where the content is spammy, and token holders only seek to maximize the number of eyeballs on the list. There are critical design decisions — such as how tokens are initially allocated, how consumers are recruited, and the non-monetary payoffs from voting for high-quality content — than can increase the probability that the good equilibrium occurs rather than the bad one.
* We’re restricting the discussion to one-shot, normal-form games of complete information. There are many more complicated types of games, and corresponding variations on Nash equilibrium. For an encyclopedic presentation of game theory, see Drew Fudenberg and Jean Tirole’s Game Theory (MIT Press, 1991).