The Horizon-Collapse Linter
Non Xero Sum · Mechanisms / Games · from Bernard's Solved Game, Issue 5
Cooperation held by threat needs a future to threaten. Put a visible endpoint on the calendar and backward induction eats the cooperation from the end forwards. This finds the round it dies.[1]
§ I — Abstract
The Folk Theorem proves cooperation can be sustained by the threat of future punishment — as long as the shadow of the future is long enough. A visible endpoint (a season-end auction, a vesting cliff, a task-loop stop condition) shrinks that shadow to a countable number of rounds. This tool walks backward from each endpoint and finds the round where the discount factor raised to the remaining rounds first drops below the Folk Theorem threshold of 0.90 — the round at which no future punishment is credible enough to keep an agent honest, and cooperation unravels. Set the numbers below and watch the collapse round move.[1]
Fig. 1 — the earliest round at which cooperation stops being rational, across all visible endpoints. The number the mechanism has to push past the game's end. It updates as you edit.
§ II — Run it
| endpoint round | name |
|---|
Nothing is fetched and nothing is stored — the check runs entirely in your browser. Point it at any game with a visible end: a token vesting cliff, a tournament's final round, an agent task-loop with a stop condition, a season-end auction. Wherever a payoff schedule has a countable last move, cooperation can unravel from it.
View source — the exact code that runs when you edit
§ III — Mechanism
The fix is not a longer game; it is a horizon with no visible floor. Rolling seasons, carried-forward reputation, and overlapping phases all do the same thing: they keep every agent uncertain about which round is the last, so the shadow of the future never collapses to a countable number. Toggle rolling horizons above and the verdict flips to safe on the same numbers — because the endpoint stops being a point the agents can induct backward from.[3]
One runnable mechanism, every week
This linter shipped with Issue 5 of Bernard's Solved Game — a weekly newsletter that ships one installable mechanism-design tool per issue. The tool you just ran is free. The next one, and the compounding flagship behind it, come with the subscription.
Read Bernard's Solved Game →§ References
- Fudenberg, D. & Maskin, E. (1986). The Folk Theorem in Repeated Games with Discounting.
- Benoit, J-P. & Krishna, V. (1985). Finitely Repeated Games.
- Axelrod, R. (1984). The Evolution of Cooperation.