Models of strategic behaviour with empirical supporthttps://measured-games-production.up.railway.app/2026-07-31T13:14:39+00:00Leif Heaneyhttps://measured-games-production.up.railway.app/vaccination.html2026-07-31T13:14:39+00:00Leif Heaney

Voluntary vaccination is modelled as a population game in which the payoff to declining is not fixed but generated by a transmission model running underneath the decision. Because the risk of infection falls as coverage rises, the two options become equally attractive at a coverage level strictly below the elimination threshold, for any positive perceived cost of vaccination. The shortfall is a property of the equilibrium rather than a coordination failure that better information would remove.

https://measured-games-production.up.railway.app/cyclic-dominance.html2026-07-31T13:14:39+00:00Leif Heaney

Three strategies related by intransitive dominance admit no best response and no resting point. Whether all three persist depends not on the payoff structure but on the spatial scale of interaction: local dispersal sustains rotating patterns in which every type is simultaneously invading somewhere and retreating elsewhere, while well-mixed populations lose two of the three types to demographic drift.

https://measured-games-production.up.railway.app/braess.html2026-07-31T13:14:39+00:00Leif Heaney

Under Wardrop equilibrium every route in use costs the same and no unused route costs less. Adding a link to a congestible network can move that equilibrium to a state in which every driver's travel time is higher, with no driver behaving irrationally at any point. The loss is a property of network topology rather than of the agents, and it is bounded above by a factor of 4/3 for affine latency functions.

https://measured-games-production.up.railway.app/minority-game.html2026-07-31T13:14:39+00:00Leif Heaney

Agents rewarded for joining the smaller of two groups cannot hold a shared correct expectation, since any widely held forecast invalidates itself. Replacing deduction with inductive strategy selection produces a system whose collective efficiency depends on a single parameter, the ratio of strategy space to population, and which shows a sharp transition between a herding regime worse than random and an efficient regime better than it.

https://measured-games-production.up.railway.app/war-of-attrition.html2026-07-31T13:14:39+00:00Leif Heaney

In a contest where both parties pay for the full duration and only the more persistent wins, no fixed commitment is stable, since any commitment is invadable by one marginally longer. The unique evolutionarily stable strategy is a negative exponential distribution of persistence times, under which expected payoff is zero for every choice and the entire value of the contested resource is dissipated.

https://measured-games-production.up.railway.app/colonel-blotto.html2026-07-31T13:14:39+00:00Leif Heaney

Two players allocate fixed budgets across several fronts, each front falling to the higher bid. No pure strategy survives, since any allocation can be countered by conceding its strongest fronts and narrowly outbidding it elsewhere. Equilibrium is a distribution over allocations with uniform marginals on the support, and asymmetric budgets produce an atom at zero for the weaker player: the underdog abandons fronts rather than spreading thin.

https://measured-games-production.up.railway.app/producer-scrounger.html2026-07-31T13:14:39+00:00Leif Heaney

Foragers choose between searching for resources and exploiting the discoveries of searchers. Because the return to exploiting falls as exploiters become common, the two payoffs cross once and the group settles at a stable interior proportion determined by the finder's share and group size alone. Free-riding is self-limiting wherever its payoff is negatively frequency dependent.

https://measured-games-production.up.railway.app/matching-pennies.html2026-07-31T13:14:39+00:00Leif Heaney

In a strictly competitive game with no pure equilibrium, unpredictability is itself the strategy, and the minimax theorem prescribes an exact mixture. Testing that prescription requires a setting where payoffs are observable, strategies are few and outcomes are immediate. Penalty kicks in professional football supply one. Across 1,417 kicks, scoring probabilities are statistically identical across strategies and choices are serially independent, making this the first natural setting in which both implications of the theorem hold. The serial independence is the harder result: in laboratory and psychological studies people switch too often, and professionals here do not.

https://measured-games-production.up.railway.app/volunteers-dilemma.html2026-07-31T13:14:39+00:00Leif Heaney

A group benefits if at least one member acts, but acting is costly and one volunteer suffices. The symmetric mixed equilibrium has each member volunteering with a probability that falls as the group grows, and the probability that nobody at all volunteers rises with group size. Diffusion of responsibility is therefore a prediction of rational play rather than a failure of it, and the effect is present in the arithmetic before any psychology is added.

https://measured-games-production.up.railway.app/common-pool.html2026-07-31T13:14:39+00:00Leif Heaney

Harvesters draw from a renewable stock whose growth depends on its own size. Each additional unit of effort yields a private return while reducing the stock available to everyone, so open access drives effort upward until rent is entirely dissipated. Because stock growth is humped rather than monotone, the resulting equilibrium can sit past the point of maximum sustainable yield, where further effort lowers the catch and collapse becomes possible.

https://measured-games-production.up.railway.app/bank-runs.html2026-07-31T13:14:39+00:00Leif Heaney

A depositor prefers to withdraw only if enough others withdraw, which under common knowledge of the bank's condition supports both a run and no run as equilibria and leaves the theory unable to say which occurs. Giving each depositor a slightly noisy private signal instead removes the multiplicity: play is uniquely determined by a threshold on the signal, and the run occurs precisely when fundamentals fall below a level the model can name.

https://measured-games-production.up.railway.app/sex-ratio.html2026-07-31T13:14:39+00:00Leif Heaney

Because every offspring has one mother and one father, the rarer sex enjoys a per-capita reproductive advantage, and selection on parental allocation drives the population to equal investment. The argument is frequency dependence applied to an allocation rather than to a behaviour. Where mating is local and brothers compete for the same mates, the same reasoning predicts a specific female bias set by the number of foundresses sharing a patch.

https://measured-games-production.up.railway.app/winners-curse.html2026-07-31T13:14:39+00:00Leif Heaney

When bidders hold noisy estimates of a common value, the winner is disproportionately the bidder whose estimate was too high. Winning is bad news about the estimate that produced it, and a bidder who does not condition on that fact overpays systematically. The correction required grows with the number of rivals, so more competition calls for lower bids, which is the reverse of the intuition most bidders bring.

https://measured-games-production.up.railway.app/ideal-free.html2026-07-31T13:14:39+00:00Leif Heaney

Foragers distributing themselves across patches of differing quality should arrive at the allocation that equalises the return in every occupied patch, which puts the share of competitors in each patch equal to that patch's share of the total input. The equilibrium requires no communication, no leader and no memory of past distributions, only that each individual can tell whether it would do better somewhere else.

https://measured-games-production.up.railway.app/virulence.html2026-07-31T13:14:39+00:00Leif Heaney

A parasite that exploits its host harder transmits better but kills faster, and the two effects trade off in the basic reproduction number. Selection therefore favours an intermediate level of harm rather than either avirulence or maximum exploitation. Where several strains share a host the calculation changes, because within-host competition rewards growth that the strain would not choose if it had the host to itself.

https://measured-games-production.up.railway.app/market-entry.html2026-07-31T13:14:39+00:00Leif Heaney

Players decide independently whether to enter a market whose capacity supports only some of them. Entry pays when few enter and loses when many do, and every allocation that exactly fills the capacity is an equilibrium, so nothing in the theory says which one occurs. What laboratory groups do instead is track the capacity closely in aggregate from the earliest rounds, while individual behaviour remains highly variable.

https://measured-games-production.up.railway.app/focal-points.html2026-07-31T13:14:39+00:00Leif Heaney

In a pure coordination game every matching pair of choices is an equilibrium and the payoffs cannot distinguish between them. What people use instead is salience: some option stands out, and the reasoning runs through what each expects the other to notice rather than through what either prefers. Coordination rates in experiments substantially exceed both chance and the rate implied by people's own preferences over the same options.

https://measured-games-production.up.railway.app/brood-parasitism.html2026-07-31T13:14:39+00:00Leif Heaney

A parasite that lays in another bird's nest and a host that can learn to spot it are locked in reciprocal selection: better mimicry favours more discriminating hosts, and more discriminating hosts favour better mimicry. The host's problem is one of signal detection rather than of incentives, because own eggs and parasite eggs produce overlapping distributions of appearance and any threshold rejects some of both. Where rejection errors are costly and parasitism is rare, accepting a good mimic is the better decision, so the arms race can rest rather than escalate without limit.

https://measured-games-production.up.railway.app/mimicry.html2026-07-31T13:14:39+00:00Leif Heaney

A palatable species that resembles a defended one is protected only while predators keep meeting the real thing. As mimics become common relative to their model, predators encounter more edible copies, learn to attack the pattern, and the protection erodes. The frequency dependence runs through a third party's learning rather than through competition between the mimics themselves, and it predicts that mimicry should fail precisely where the model is scarce.

https://measured-games-production.up.railway.app/rent-seeking.html2026-07-31T13:14:39+00:00Leif Heaney

Competitors spend to raise their probability of winning a prize rather than to win it outright, which makes effort an investment in odds. Equilibrium spending rises with the number of competitors and approaches the whole value of the prize, so the resources devoted to capturing it approach the resources at stake. In the laboratory people reliably spend more than that, dissipating more than the prize is worth, and the overspending is one of the most replicated findings in experimental economics.