Guess 2/3 o the Mean

• Yer level shawes hou mony steps forrit ye thocht. A guess near 50 is level 0: nae thocht aboot the ithers. Near 33 is level 1: "fowk will guess aboot 50". Near 22 is level 2: "fowk will think ane step forrit". Ilka step is 2/3 o the yin afore it [1].
• Game theory's answer is 0. Gin ilkane thocht it richt oot, ilka guess wad shrink tae 0. Real players never dae this, sae 0 has never won a public contest [1][2].
• Can never win: a guess abuin 66.7 cannae win, acause even gin ilkane guessed 100, the target wad be 66.7.
• This week's winner appears whan the round ends (ilka round is a week, Monday tae Sunday, UTC). We never shaw the live mean afore syne: it wad spoil the gemme. A round needs at least 30 players tae hae a result.

1. Pick a hale nummer fae 0 tae 100 and lock it in.
2. Ilkane's guesses this week are averaged whan the round closes, and the target is 2/3 o that mean.
3. The nearest guess wins. Ties share the win. On Monday, the page shawes last week's players, mean, target, winnin guess and hou ilk nummer wis guessed.

Ane guess per body counts in ilk round. We count ane guess per browser and ane per internet connection, sae fowk sharin a network (a schuil, an office, a faimly) count ance a week. A second guess disnae count. Richt efter ye guess, the page shawes whaur yer nummer sits on the reasoning ladder and hou it wad hae fared in the weel-kent contests abuin.

Economists caa this gemme the p-beauty contest, wi p = 2/3 here. Rosemarie Nagel made it a weel-kent lab experiment in 1995 [1]. Her findin: fowk dinnae reason richt doon tae 0, and they dinnae guess at random either. Maist stop efter ane or twa steps o "they think that I think".

That idea, that fowk reason a limited nummer o steps aboot ithers, became the level-k and cognitive hierarchy models o strategic thinkin [4][7]. The newspaper contests that cam efter, wi thoosands o readers ilkane, shawed the same steps at 33 and 22, plus a group that chose 0 or 1 [2].

Yer level describes ane guess in ane gemme. It isnae a measure o intelligence, and a deeper level isnae better: the winner is wha predicts the crowd best, and the crowd uisually stops at ane or twa steps. A level-4 guess is clever reasoning that uisually ends up ablo the target.

The players here are whaever visits that week, no a fixed sample, sae the target moves fae week tae week. The weel-kent contests haed slightly different rules: the Financial Times allooed hale nummers anerly, and Politiken and Spektrum allooed decimals [2][3].