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 the future shrinks as it approaches. Give this your payoffs and it finds the round cooperation stops being sustainable; give it none and it shows you the answer for every payoff ratio.[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 computes the value of the rounds still to come. Cooperation holds while that value covers the temptation-to-punishment ratio your payoffs imply: (T−R)/(R−P). There is no universal threshold; it comes from the payoffs. Leave the payoffs blank and the tool names no round and shows the collapse round for every ratio. The final round always fails, because nothing follows it.[1]
Fig. 1 — with payoffs, the earliest round at which cooperation stops being sustainable, across all visible endpoints; without them, no round (see the table). 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.
- Selten, R. (1978). The Chain Store Paradox; Luce, R. D. & Raiffa, H. (1957). Games and Decisions. Backward unravelling when the stage game has a single equilibrium.
- Benoit, J-P. & Krishna, V. (1985). Finitely Repeated Games. With several stage-game equilibria, cooperation can survive a known endpoint, and this tool's flag does not apply.
- Axelrod, R. (1984). The Evolution of Cooperation.