Cryptoeconomic Theory: Pareto Efficiency

Foundations of Game Theory and Cryptoeconomics

This is part 3 of the weekly Cryptoeconomics series

Let’s say you’re designing a new set of protocols for a blockchain-based organization. If you’re attempting to be helpful, you’d ask questions along the lines of: “How do we design the interaction of individuals that leads to an aggregate outcome that is “socially good”?

Unfortunately that doesn’t get anyone far at all. What does “good” even mean? How are two protocols compared to see which is “better”? And further, what does “better” even mean?

Efficient Outcomes

One criterion for evaluating the outcome of a social interaction is that it should be efficient. More specifically– resources in an economic state should be allocated in the most efficient manner, every time, no matter the interaction.

There are two things we want from an ideal outcome:

  1. An outcome maximizes total payoff (total surplus) over all other sets of feasible outcomes.
  2. An outcome is preferred by an individual (economic agent) over all other sets of feasible outcomes.
Notice how every condition refers to a relative comparison between two outcomes. The potential movement from potential outcome A to potential outcome B that provides the most practical value is the core basis of all game theory.

A transition from outcome A to B that makes someone better off (a winner) without making anybody else worse off (a loser) is a Pareto improvement (PI).

An outcome is called Pareto Efficient when there is no possibility for Pareto improvement, i.e. there exists no other feasible outcome (using available resources and technologies) that would make at least one person better off without making anyone worse off.

To simplify: there are winners and losers when moving from outcome A to outcome B. The concept of Pareto Efficiency states that even though there are some losers, the winners can compensate the losers, and thus the outcome is “socially good”.

The table to the left gives the utility of each individual 1,2 and 3 for each of the feasible alternatives A, B, C, D, and E.

In plain english, if these 3 individuals do something (A,B,C…), they get a certain amount of satisfaction (utility).

Pareto Efficiency is determined by testing Pareto improvement of the many potential possibilities. B is Pareto efficient- from B → C, 3rd individual goes from 60 → 50. C is Pareto efficient- 25 and 50 are max utilities for individual 1 and 2 respectively. D is also Pareto efficient- from D → C, 3rd individual goes from 70 → 50. From D → B, the 3rd individual goes from 70 → 60. From A → C everyone stays the same or increases so A is not Pareto efficient. From E → D, everyone stays the same or increases so E is not Pareto efficient.

Imagine another simple economic problem where you, me, and Sarah come across 6 free bitcoin.

The amount each of us gets in various scenarios looks like this:

Total surplus is a fancy way of counting how many bitcoin we have total.

A=6 , B=6, C=5 , D=6, E=6.

All decisions that have 6 are Pareto Optimal and are the maximum out of the group. There is no way to go to another option and have more.

Also notice how in option A, you receive all 6 bitcoin leaving everyone else with 0. This is still Pareto Optimal and “socially good” but isn’t “fair”. Sarah and I would not enjoy such a system. This tug and pull between fairness and efficiency has historically been a major point of discussion.

An anonymous Quora user simplified these concepts of Pareto Efficiency best:

My mom packed me a ham sandwich for lunch but I don’t like ham; my friend’s mom packed her a turkey sandwich for lunch, and she likes turkey and ham sandwiches equally well. Nobody else cares what we eat for lunch.
My friend and I trade sandwiches; I am better off than I was before, and nobody is worse off. This is an example of a Pareto improvement.

Intro to Game Theory

Consider negotiation between two persons in two situations. Suppose each negotiation has four outcomes a, b, c and d. For the first situation the payoff to player i (Πi) for each outcome is: a ( Π1 = 1, Π2 = 2), b ( Π1 = 3, Π2 = 3), c ( Π1 = 4, Π2 = 5), d ( Π1 = 6, Π2 = 7). For the second situation the payoff to player i (Πi) for each outcome is: a ( Π1 = 6, Π2 = 2), b ( Π1 = 4, Π2 = 4), c ( Π1 = 3, Π2 = 5), d ( Π1 = 1, Π2 = 7).

For each situation we drew a graph of outcomes with payoff to player 1 on the vertical axis and the payoff to player 2 on the horizontal axis.

We will answer:

  1. Which outcomes are Pareto Efficient?
  2. How does this relate to Social Order (cooperation and coordination)?
  3. ‘Common interest’ game vs Conflict of interest’ game?

Situation 1

a=3 , b=6 , c=9 , d=13

When calculating total payoffs for each outcome, only (d) is Pareto Efficient.

Notice how the graph trends up and to the right, with each outcome increasing total surplus relative to the last; this is called “Pareto Ranked”.

When evaluating outcomes, would these players cooperate together to reach (d)? Yes. We call this a Common Interest Game. A traffic jam is an example of common interest: jams are generally poor outcomes, and managing to avoid them would benefit everyone.

Out of the two problems for social order —

  1. Lack of Cooperation
  2. Lack of Coordination

–– there is indeed cooperation, and only a lack of coordination is present in this example.

In the real world this is not common, there typically also exists a cooperation problem, or extreme cooperation problems like in situation 2.

Situation 2

a=8 , b=8 , c=8 , d=8

When calculating total payoffs for each outcome in this example, all are Pareto Efficient.

When evaluating outcomes and the players, we call this a Pure Conflict Game. In our Bitcoin sharing example above, payoffs such as (6,0,0) or (2,0,4) have very disproportionate personal gains. You, me, and Sarah could potentially get into a fist fight to decide distribution of our Bitcoin.

Out of the two problems for social order —

  1. Lack of Cooperation
  2. Lack of Coordination

— there exists both a cooperation AND a coordination problem.

Solving social order here is then much more difficult.

In practical applications, there is normally a mixture between conflict and common interest.

Here, out of the many possible outcomes, transitioning from c → b is only a coordination problem, while c → a and c → d has both problems. We’ll use this later to understand why the Tragedy of Commons exists; how agreements to cooperate are monitored and enforced, and further define cooperative vs non-cooperative games.

Read the next post. Read the previous post.

Edited by: Steven McKie