Decay functions: how a rating forgets last season

Forgetting as a design requirement

A rating that never forgets becomes a career total, which is a legitimate quantity and useless for describing how strong a competitor is now. A rating that forgets immediately becomes a description of the last event, which is precise about the present and blind to consistency. Every practical system sits between those extremes, and the position it occupies is set by a decay function applied to older results.

That function is rarely explained to readers, yet it determines whether the table behaves as a form guide or as a seasonal summary. Two systems identical in every other respect will produce visibly different orders purely through their choice of decay.

Common shapes and their behaviour

A step function counts everything inside a window fully and nothing outside it, which is simple to explain and produces abrupt jumps at expiry. A linear taper reduces weight steadily and removes those jumps, at the cost of a rule that is harder to describe in a sentence. An exponential decay reduces weight by a constant proportion per period, which is mathematically convenient and never fully discards anything.

Each shape produces a characteristic pathology, whether that is cliff edges, a long tail of stale influence or excessive sensitivity to the newest result. Choosing among them is a question about which failure mode is most tolerable for the intended use.

Matching decay to the rate of real change

The principled way to set decay is to ask how quickly the underlying quantity genuinely changes, and to forget at roughly that rate. Ability in a physically demanding discipline changes over months and years rather than weeks, which argues for slower forgetting than most tables use. Circumstances change faster, since a competitor role, conditions and supporting cast can alter within a single season.

A rating that mixes ability and circumstance therefore has no single correct decay, because its components change at different speeds. Separating them, where the data allows, is more useful than tuning a compromise setting that suits neither.

Decay and the incentives it creates

Fast decay rewards recent activity and encourages competing frequently, since points earned long ago provide little protection. Slow decay rewards a strong period and allows a competitor to remain highly placed through a quiet stretch, which can look like stagnation. Neither incentive is inherently better, and the appropriate one depends on whether the sport wants to encourage participation or to reward sustained peaks.

Systems used for entry and seeding usually favour faster decay, because access should track current competitiveness rather than past standing. Systems used for awards usually favour slower decay, because an award is meant to recognise a body of work.

Explaining decay to readers

Most disputes about a rating being wrong are actually disputes about decay, and they would evaporate if the setting were stated openly. A single sentence describing how quickly results lose weight tells a reader more about the table than a full description of the scoring formula. It also lets them predict how the table will behave, which builds trust far more effectively than assertions of accuracy.

Where a system publishes its decay, criticism tends to become technical and specific rather than general and hostile. That shift is worth more to a governing body than any improvement in the formula itself.