Feasible and Individually Rational Payoffs

1. The Set of Feasible Payoffs: What is Possible?

The most obvious payoffs a pair of players can achieve are those resulting from the pure strategy profiles of the stage game. Let's use the Prisoner's Dilemma from one of our key resources as a working example.

The pure strategy payoffs are (2,2), (0,3), (3,0), and (1,1). If we plot these on a graph with Player 1's utility ($u_1$) on the x-axis and Player 2's utility ($u_2$) on the y-axis, we get four distinct points.

But can players achieve payoffs between these points? Yes. By using a shared randomizing device (like a virtual coin flip or die roll that they both observe), players can coordinate their strategies. For example, they could agree to play (Defect, Cooperate) if a coin is heads and (Cooperate, Defect) if it's tails. If the coin is fair, the expected payoff would be:

By varying the probabilities, they can trace out the entire line segment between (3,0) and (0,3). By extending this logic, players can achieve any weighted average of the pure strategy payoffs.

This brings us to a core concept: the set of all achievable average payoffs is the convex hull of the pure-strategy payoff vectors. A convex hull is the smallest convex set that contains all the points. Visually, it's like stretching a rubber band around the outermost points.

For our Prisoner's Dilemma example, the feasible set is the diamond-shaped area formed by connecting the points (3,0), (2,2), (0,3), and (1,1).

To formalize this and see it illustrated, I recommend two resources. First, a formal definition, and second, a clear graphical explanation.

In summary, the feasible set represents every possible average payoff profile the players can engineer through coordinated play, including randomization. Any payoff outside this set is physically impossible to achieve.

2. The Set of Individually Rational Payoffs: What is Acceptable?

Just because a payoff is feasible doesn't mean a rational player would agree to a strategy that yields it. A rational player will not accept an outcome if they can get a better payoff on their own, regardless of what others do. This introduces the idea of a "security level" or a minimum acceptable payoff.

This security level is called the min-max value. For a player $i$, their min-max value, denoted ${\underline{v}}_i$, is the payoff they can guarantee for themselves, even if all other players collude to try and minimize player $i$'s payoff.

Formally, for a two-player game, player 1's min-max value is:

This expression means: we first find player 1's best response (${\max }_{{\alpha }_1}$) for every possible strategy player 2 might play. Then, we find the worst-case scenario for player 1, where player 2 chooses the strategy (${\min }_{{\alpha }_2}$) that makes player 1's best-response payoff as low as possible. That minimum possible "best-response" payoff is what player 1 can always guarantee.

A crucial theorem in game theory states that any payoff a player receives in any Nash Equilibrium of the game must be at least their min-max value. A player can't get less than their security level in an equilibrium.

This gives us our second key definition: a payoff vector $v=(v_1,v_2,\dots ,v_n)$ is individually rational (IR) if every player's payoff is at least their min-max value.

The following resources will help you solidify this concept.

A common and useful simplification, especially in games with a unique stage-game Nash Equilibrium, is to use the NE payoff as the IR threshold. Since the NE payoff is guaranteed to be $\ge$ the min-max value, it serves as a valid (though sometimes not the tightest) floor for what is considered rational. In our running Prisoner's Dilemma example, the unique NE is (Defect, Defect) with payoff (1,1). The min-max value for each player is also 1 (by playing Defect, you guarantee you get at least 1). Therefore, the set of individually rational payoffs is all pairs $(u_1,u_2)$ such that $u_1\ge 1$ and $u_2\ge 1$.

3. Combining Feasibility and Rationality

We now have the two pieces we need. The interesting outcomes in a repeated game are those that are both possible to achieve and acceptable to all players. This is the set of Feasible and Individually Rational (FIR) payoffs. It is simply the intersection of the feasible set and the individually rational set.

Let's plot this for our Prisoner's Dilemma example:

1. Feasible Set: The diamond-shaped convex hull connecting (3,0), (2,2), (0,3), and (1,1).
2. Individually Rational Set: The region where $u_1\ge 1$ and $u_2\ge 1$. This is an unbounded area to the "northeast" of the point (1,1).

The FIR set is the part of the diamond that lies within this northeast quadrant.

To see the process of constructing the FIR set from scratch, the following resource is excellent. It walks through the same logic we just did, with clear diagrams for each step.

The key insight here is that the FIR set contains payoffs—like the cooperative outcome (2,2) in our example—that are strictly better for both players than the stage-game Nash Equilibrium (1,1). The possibility of achieving these superior outcomes is the central promise of repeated interaction.

Conclusion

In this lesson, we established the boundaries for cooperation and conflict in repeated games. We moved from analyzing specific strategies to defining the entire space of relevant outcomes.

Key Takeaways:

• Set of Feasible Payoffs: This is the convex hull of the pure strategy payoff vectors from the stage game. It represents all average payoffs that are physically achievable through coordination and randomization.
• Min-max Value (${\underline{v}}_i$): This is a player's security-level payoff—the minimum they can guarantee themselves regardless of what others do.
• Set of Individually Rational (IR) Payoffs: This set includes all payoff profiles where each player receives at least their min-max value. A rational player will not accept any long-term outcome below this threshold.
• Set of Feasible, Individually Rational (FIR) Payoffs: This is the intersection of the feasible and IR sets. It represents the universe of outcomes that are both achievable and strategically acceptable as long-term results.

You now possess the geometric and conceptual tools to understand one of the most powerful ideas in game theory. The FIR set defines the battleground. In our next lesson, we will introduce the Folk Theorem, a remarkable result which states that, under the right conditions, any point within this FIR set can be sustained as a stable equilibrium outcome.