Tensions in Practice: Invasion resistance does not maximize the group¶
Two conventions in one declared encounter game
In an invented symmetric encounter game, A earns 4 against A and 0 against B; B earns 3 against A and 2 against B. Let p be the population share using A. Under random mixing, A’s expected payoff is 4p and B’s is 2+p. Both uniform conventions resist a rare alternative: against all A, 4 beats 3; against all B, 2 beats 0. Yet all A earns 4 per member and all B only 2. Their attraction regions also differ under the stipulated rule that the higher-payoff type grows.
Obtain the higher coordinated payoff
Establish and sustain the all-A convention with payoff 4.
Retain the wider local attraction region
Use the all-B convention, which recovers from a larger share of the alternative in this model.
Why these aims pull against each other
Both conventions pass the small-invasion test. A offers higher collective payoff but requires A’s share to stay above two thirds for the declared growth rule to favor it; B is favored below that boundary.
Choose an arrangement to see what changes and what remains difficult.
p is the share using A. Expected payoffs are A = 4p and B = 2+p; Favored names the type with greater payoff. 10% other means 10% of the nonresident type. 8/3 is 2⅔. Each row is a separate population composition, not a time step.
What this choice protects
What it costs
When it fits
Compare the arrangements
Establish convention A
Start with all A, then inspect alternative population compositions using the same payoff matrix. A remains favored with 10% B, but loses its advantage when its share falls to 60%.
| A payoff | B payoff | Favored | |
|---|---|---|---|
| All A | 4 | 3 | A |
| 10% other | 3.6 | 2.9 | A |
| A = 2/3 | 8/3 | 8/3 | Tie |
| A = 60% | 2.4 | 2.6 | B |
- What it protects
- The uniform population receives payoff 4, the larger of the two uniform-convention payoffs.
- What it costs
- The attraction region is narrower: A must exceed two thirds. Establishing or restoring that share can require coordinated change rather than isolated switches.
- When it fits
- Plausible when the higher coordinated payoff is worth the establishment burden and the population can remain within the stated basin.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
Retain convention B
Start with all B. B remains favored after 10% A and continues to be favored until A exceeds two thirds; at 70% A, selection favors A instead.
| A payoff | B payoff | Favored | |
|---|---|---|---|
| All B | 0 | 2 | B |
| 10% other | 0.4 | 2.1 | B |
| A = 2/3 | 8/3 | 8/3 | Tie |
| A = 70% | 2.8 | 2.7 | A |
- What it protects
- B has the wider attraction interval under the declared growth rule and can recover from a larger fraction of A.
- What it costs
- The uniform population earns only 2, despite the feasible all-A outcome yielding 4 to everyone. Invasion resistance does not certify collective optimality.
- When it fits
- Plausible when remaining robust within this wider basin is worth forgoing the higher uniform payoff or coordinated transition is not feasible.
Illustration note: This is an editorial, deliberately bounded illustration. Its stated rules and any numbers are invented, not observations, recommended settings, or predictions.
What this illustration does—and does not—establish
The source establishes the structural tension; the concrete alternatives and their conditional costs are editorial synthesis. No arrangement is a universal recommendation.
- Payoffs and random mixing are invented. A continuous replicator-style sign rule is stipulated for the basin discussion; the table alone is not an empirical population forecast.
- The p = 2/3 interior tie is not a robust resident state. Small deviations to either side are favored away from it under the declared rule.
- Foresighted people can deliberately coordinate a switch; they need not follow blind selection dynamics. Only these two strategies and fixed payoffs are tested.
Source entries
Evolutionarily Stable Strategy
Evolutionarily stable strategy Stable versus optimal (sign/direction) supplies the local tension. The setting, alternative arrangements, and stipulated consequences are editorial applications.
Stable versus optimal (sign/direction)
Diagnostic: ask whether the resident maximizes group payoff or merely resists deviation; stability is about perturbation-survival, not optimality, and a Pareto-dominated outcome can be a perfectly solid ESS.
Local stability versus basin size (scalar)
The invasion test certifies resistance to *small* mutant injections, but says nothing about how large a coordinated shock is needed to escape — two ESSs equally "stable" by the test can have wildly different basin sizes.