Beyond Cycles of Deviation: How Fair Principles Ensure a Stable Nash Equilibrium

Table of Links

Abstract and 1. Introduction

2. A free and fair economy: definition, existence and uniqueness

   2.1 A free economy

   2.2 A free and fair economy

3. Equilibrium existence in a free and fair economy

   3.1 A free and fair economy as a strategic form game

   3.2 Existence of an equilibrium

4. Equilibrium efficiency in a free and fair economy

5. A free economy with social justice and inclusion

   5.1 Equilibrium existence and efficiency in a free economy with social justice

   5.2 Choosing a reference point to achieve equilibrium efficiency

6. Some applications

   6.1 Teamwork: surplus distribution in a firm

   6.2 Contagion and self-enforcing lockdown in a networked economy

   6.3 Bias in academic publishing

   6.4 Exchange economies

7. Contributions to the closely related literature

8. Conclusion and References

Appendix

3.2 Existence of an equilibrium

In this section, we state and prove our main result.

In the strategic form game in Table 4, the sum of excess payoffs in any cycle of outcomes equals 0. Therefore, the game does not admit a cycle of deviations. The profile x ∗ = (a2, b3) is the only pure strategy Nash equilibrium of the game.

Note that the game in Table 4 is generated from a free and fair economy. From Definition 7, a sufficient condition for a finite strategic form game to admit a pure strategy Nash equilibrium is the absence of a cycle of deviations. The sum of excess payoffs in any cycle of deviations has to be strictly positive, as illustrated in Table 3 in Example 2. Such an example of a cycle of deviations can not be constructed in a strategic form game generated from a free and fair economy (see Table 4 in Example 2). We prove that in a strategic form game generated by a free and fair economy, the sum of excess payoffs in any cycle of deviations equals 0.

The principles of market justice that define a free and fair economy are only sufficient conditions for the existence of a pure strategy Nash equilibrium. However, an economy that violates the fair principles may not have a pure strategy Nash equilibrium.