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Values Game Theory

Short Explanation

Game Theory

- Players are most likely to stake when they anticipate an expansion in supply and/or price.
- Players are most likely to sell when they anticipate a contraction in supply and/or price.
- Players are most likely to bond when they do not have a strong directional bias but donβt anticipate significant downside.
- Game Theory is the study of mathematical models which can be used to determine the advantage or disadvantage of specific choices in a given scenario.

Explanation

- 4,4 and 3,3 are some of the labels with give to certain actions with regards to Game Theory.
- In the case of Values, we use it to describe the benefits and drawbacks of the various ways in which users can interact with the Protocol.
- Below is a the Values Game Theory Matrix. It is a grid which you can use to determine the benefit of users taking actions within the Protocol.
- The columns indicate your actions while the rows indicate the actions of others. Each action is assigned a value which translates to outcome values on the table.
- For example:
- if both parties stake, the outcome value is 3 + 3 = 6. This is considered a highly positive outcome for the user and the Protocol.
- If one user stakes whilst another buys a 4,4 bond, the outcome value is 7 (3 + 4). This is also considered a highly positive outcome.
- But if one user sells whilst the other stakes, this is an unfavorable outcome (-1 + 1 = 0). If both users sell, the combined downward price movement negatively affects them both.
- Therefore, an ideal scenario would be for all users to Stake and buy 4,4 Bonds. This allows the Protocol to build reserves and offer more attractive APYs whilst also allowing users to increase their stake and reduce their cost basis.

Introduction

βValues is an innovation in the way people interact with financial protocols. The motivation behind this essay is to delve deep into the Values protocol, not from an operational or financial perspective, but with the purpose of analysing human interactions. There are many fields that have something to say about human interaction but the field of study we will focus on here will be game theory - the study of strategic interdependence. It is our belief that Values is solving the problem of creating a new currency through internal coordination between different stakeholders within the protocol, without resorting to any policy enforced by a central entity. At its core this is an example of a prisoner's dilemma. A prisoner's dilemma is a situation where an individual's self interest is in conflict with a common goal, leading to the players within the game not cooperating despite it being in their best interests to cooperate. In the case of currency, it is in each individual's best interest to use the most liquid, most widely used & stable currencies. In crypto, the assets that best meet these requirements have been dollar denominated stablecoins. The βcommon goalβ here is to remove reliance on centralized fiat currency and to maintain the individual's purchasing power. These are goals which stablecoins categorically do not achieve and yet still they are some of the most used assets in crypto. No individual acting alone can disentangle crypto from fiat; It takes mass coordination, along with alignment of incentives, and this is what Values facilitates. We will begin by outlining the essentials of game theory and analysing the prisoner's dilemma from a purely abstract perspective. We will then dive into the specific components of Values. We will take no shortcuts. Values is a complex protocol that is the first of its kind and is deserving of deep and thorough analysis. This is a collaborative piece of work that will be revised over time. It is aimed at a broad range of readers and if you would like to offer feedback or contribute to future sections, please join the Values DAO discord and reach out to the Community and Content team.

Game Theory Essentials

What is Game Theory?

Game theory is the study of strategic interdependence. There are many situations in life where the best response to a situation depends on what other people in that situation do. Strategic situations are interesting because we can often get multiple outcomes, some of which are remarkably stable and yet sub-optimal for all people involved. Consider a macroeconomic application of multiple stable outcomes under these conditions:

- Firms only invest if customers will buy goods.
- Customers only buy goods if they are paid wages.
- Customers are paid wages by firms only if they buy goods.

The optimistic (non-exhaustive) outcome:

- Firms think customers will buy goods and so expand production.
- Customers buy these goods because they think their wages are secure.
- The expanded profits allow firms to pay higher wages.

The pessimistic outcome:

- Firms do not think customers will buy goods and so restrict production to save costs.
- Customers restrict their purchasing because they lack wage security.
- Firms profits fall, leading to cost cutting in the form of wage cuts and layoffs.

We can see that no one wants to be in the pessimistic outcome and yet it is plausible. To understand why, we use game theory.

The Model

In essence, a game theoretic model has the following components: **Players.**

They make choices based on information they hold about themselves, the other players, and the structure of the game: **Strategy.**

The full set of choices a player makes in a game: **Payoffs** .

A payoff is the reward to each player, contingent upon the outcome of their interactions. We decide payoffs by considering each player's preferences within the game: **Payoff Matrix.**

A table that lists all the available strategies and their respective payoffs. Game theory is useful because it allows us to determine the optimal strategy which produces the best outcome for all the players involved. The best way to get to grips with the power of game theory, is with an example.

The Prisoner's Dilemma - A Simultaneous Game

The first game any student of game theory learns is the prisoner's dilemma. This is due to the fact that it is a simple game with applications to a wide variety of strategic situations. Once you see and understand it, you will see it at play everywhere.

Context

The story goes like this. Two thieves plan to rob a store. As they approach the door, the police arrest them for trespassing. The police suspect that the pair planned to rob the store but they lack the evidence to prove it. They therefore, require a confession to charge the suspects with the more serious crime. The interrogator separates the suspects and tells them each: *βWe are charging you with trespassing which will land you in jail for a month. I know you were planning to rob the store but I cannot prove it without your testimony. Confess to me now, and I will dismiss your trespassing charge and set you free. Your friend will be charged for the attempted robbery and face 12 months in jail. Iβm offering your friend the same deal. If you both confess, your individual testimony is no longer valuable and both of you will receive 8 months in jail.β *Both players are self-interested and want to minimise their jail time. What should they do?

Payoff Matrix

Using a payoff matrix allows us to condense all the information into an easy-to-analyse diagram:

Player 1βs available strategies are the rows (Quiet or Confess) and their corresponding payoffs are the first numbers in each cell.Players 2βs available strategies are the columns and their corresponding payoffs are the second numbers in the cells.If player 1 stays Quiet and player 2 stays Quiet the game ends in the top left corner of the matrix. If both players Confess the game ends in the bottom right corner of the matrix and so on.

Strategies

To see which strategy each player will choose we should look at each move in isolation. From player 1βs perspective, what should he do if he thinks player 2 will stay Quiet?

We can see that player 1 should Confess because if he stays Quiet he will get one month in jail and we have already stated that both players prefer less time in jail.

What about if player 1 thought that player 2 was going to Confess?

Again it seems that player 1 should Confess as it leads to 8 months jail time rather than the 12 he will receive for keeping quiet.

Putting this together we reach an important conclusion: **Player 1 is better off confessing regardless of player 2βs strategy. **

Letβs look at player 2βs perspective, assuming he thinks player 1 will stay quiet:

Looking at the second numbers now we can see that like player 1, player 2 should confess as well: he will be set free instead of getting 1 month in jail.

Once again it looks like Confess should be the chosen strategy even if player 2 thinks player 1 will also Confess.

Assumptions and Conclusions

- We assumed that both players' preferences were to minimise their jail time
- We assumed both players were self-interested (i.e they don't care about their friendsβ fate)
- We assumed only one interaction
- We assumed the players could not interact and plan their responses in advance

These assumptions led to a sub-optimal outcome in the game (Confess, Confess). We can see that had both players stayed quiet, they would have received less jail time. This is an **unstable equilibrium** because βas we saw β both players are motivated to confess if they believe the other will stay quiet. Confess, Confess is therefore the only Nash equilibrium. A Nash equilibrium is a state in a game where no player wishes to deviate from their strategy, given what the other players are doing. **If both players were able to cooperate with each other and stay Quiet however, they would have achieved a better outcome. This is an important conclusion as it shows us that two individuals may not cooperate, despite it appearing to be the best strategy for both.**

Real World Example

We can apply the model of the prisoner's dilemma to many real world situations. For example imagine the players being replaced with two countries. The strategies available to both countries are to arm themselves with weapons (replacing Confess) or disarm (replacing Quiet). In our first example the payoffs represented jail time. Payoffs do not actually have to represent a βrealβ thing, they are really just the preferences and assumptions of the players captured in a number. What is important is that the payoffs have meaning relative to each other, and they remain consistent. So in our first game both players preferred lower scores. If we continue to assume that both players (countries) in this game want to minimise their score we get the following matrix:

Following the same steps as in the original game we can see that unfortunately (Arm, Arm) is the only dominant strategy. It is for this reason that efforts to unilaterally disarm are (in this author's opinion) doomed to fail. The specific payoffs chosen do not matter. What matters is that the relative differences between the payoffs fit our assumptions. Game theory cannot predict the future, it can only illustrate outcomes based on the preferences we set (via the payoffs chosen).

The Iterated Prisoner's Dilemma

In our first example of the prisoner's dilemma, we assumed only one interaction. What changes if we play the game over and over again?The single interaction analysed in our first example is not particularly realistic. Life is full of repeated interactions and in the case of Olympus, if it is to achieve its goal of becoming a reserve currency, it will take many repeated interactions.For this game we will move away from the story of the prisoners dilemma and focus instead on the payoffs. The best outcome for either player is the highest number. Strategies have also changed to cooperate and defect as these are broader terms that better reflect a wide range of real life situations.

Payoff Matrix

If you Defect and the other player Cooperates, this will result in the highest payoff the first time. However, continued defection is likely to be met with defection which shifts the players to the bottom right cell - a suboptimal outcome. Remember that the specific numbers chosen are unimportant but the relative differences between them are important. Here we are assuming that defection yields a higher payoff than cooperation. We assume this to fulfill the criteria of a prisoner's dilemma - where an individual's self interest is in conflict with a common goal. An individual is motivated to Defect. However the best possible outcomes over the course of the whole series of games occur when both players **Cooperate continuously**. For proof of this, and if you are interested in playing around with assumptions yourself, please visit https://ncase.me/trust/.This is a crucial finding of the project led by Robert Axelrod. Axelrod ran a tournament where entrants submitted computer programs that played repeated games of the prisoner's dilemma against each other. The payoffs and available strategies in this tournament were identical to those illustrated above. A total of 14 teams entered and each team programmed their entry with some rule set to select the cooperate or defect strategy each move. The tournament was structured as a round robin, so each team played every other team, and each game had 200 moves. Entrants were also paired with their own twin and a RANDOM program which randomly chose to cooperate or defect with equal probability. The tournament was repeated 5 times resulting in 120,000 moves and 240,000 separate choices.The scores from those games were tallied up, giving each entrant a final score. The aim was to score as highly as possible after all games had taken place. The winning program was called TIT-FOR -TAT and its strategy was simple: **On the first move, cooperate. Then play whichever strategy the opponent played in the previous round. **The players move simultaneously. So if the opponent defects on the first round, TIT-FOR-TAT would defect on the second. In this way TIT-FOR-TAT can't be easily taken advantage of by other programs repeatedly defecting. Defection will be met with defection.It also pays to cooperate with TIT-FOR-TAT, as it will always cooperate in return. Whilst TIT-FOR-TAT can by definition never score higher than its opponent in any one game (because it only ever matches strategies and therefore scores) it still **did better than all other entrants across the whole series of games in the tournament.**

Implications of Cooperation being a Dominant Strategy

The outcome of the iterated prisoner's dilemma tournament has meaningful implications both for wider society and for Values. We are often taught that in a capitalist economy, individuals are only concerned with their own self-interest, and therefore selfish and competitive behaviour is the norm. Axelrodβs tournament suggests that cooperation is actually the best way to win. For a full analysis of the tournament and its implications I recommend reading Axelrod's book, βThe Evolution of Cooperationβ. In the following section I will summarise the key components that are, according to Axelrod, necessary for cooperation to develop in decentralized systems.

Promoting Cooperation

How do you establish a system of cooperation amongst strangers with vastly different preferences, who are separated by time and geography, who have never met, and likely never will?If you pause to consider this question carefully, you will realise it is a problem humanity has faced since time immemorial. If there is one thing humanity is good at, it is solving problems and solving the problem of cooperation has powered human prosperity and facilitated our ingenuity like no other creature in the universe. So how have we learned to cooperate?

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Outline

Short Explanation

Introduction

Game Theory Essentials

What is Game Theory?

The Model

The Prisoner's Dilemma - A Simultaneous Game

Context

Payoff Matrix

Strategies

Assumptions and Conclusions

Real World Example

The Iterated Prisoner's Dilemma

Payoff Matrix

Implications of Cooperation being a Dominant Strategy

Promoting Cooperation