Ludometrics: luck, and how to measure it

Daniel E. Gilbert; Martin T. Wells

doi:10.1515/jqas-2018-0103

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Ludometrics: luck, and how to measure it

Daniel E. Gilbert und Martin T. Wells

Veröffentlicht/Copyright: 1. Juni 2019

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Aus der Zeitschrift Journal of Quantitative Analysis in Sports Band 15 Heft 3

Abstract

Game theory is the study of tractable games which may be used to model more complex systems. Board games, video games and sports, however, are intractable by design, so “ludological” theories about these games as complex phenomena should be grounded in empiricism. A first “ludometric” concern is the empirical measurement of the amount of luck in various games. We argue against a narrow view of luck which includes only factors outside any player’s control, and advocate for a holistic definition of luck as complementary to the variation in effective skill within a population of players. We introduce two metrics for luck in a game for a given population – one information theoretical, and one Bayesian, and discuss the estimation of these metrics using sparse, high-dimensional regression techniques. Finally, we apply these techniques to compare the amount of luck between various professional sports, between Chess and Go, and between two hobby board games: Race for the Galaxy and Seasons.

Keywords: Bradley-Terry model; game theory; generalized linear models; luck; ludology; skill

Funding source: National Institutes of Health

Award Identifier / Grant number: U19 AI111143

Funding statement: National Institutes of Health, Funder Id: http://dx.doi.org/10.13039/100000002, Grant Number: U19 AI111143. Division of Mathematical Sciences, Funder Id: http://dx.doi.org/10.13039/100000121, Grant Number: 1611893.

A Newton-Raphson updates for regularized skill estimation

For Algorithm 18, we have

lλ(o;s,t)=∑i=1nlogP(Oi=oi)−λ||s||22=∑i=1nli−λ||s||22,

where

li={log(1−Φ(t−xi⋅⊤s2))if oi=1log(Φ(t−xi⋅⊤s2)−Φ(−t−xi⋅⊤s2))if oi=0log(Φ(−t−xi⋅⊤s2))if oi=−1.

The Newton-Raphson algorithm for optimizing this loss function is:

[st]j+1=[st]j−(∇2(lλ(o;s,t)))−1∇(lλ(o;s,t))

where the above gradient and Hessian are given by

∇(lλ(o;s,t))=∑i=1n∇io−λs∇2(lλ(o;s,t))=∑i=1n(∇io)2−λ𝕀A

where

∇i1=S1(ηi1)[xi⋅⊤−1](∇i1)2=(ηi1−S1(ηi1))S1(ηi1)[xi⋅−1][xi⋅⊤−1]∇i−1=−S2(ηi2)[xi⋅1](∇i−1)2=−(ηi2+S2(ηi2))S2(ηi2)[xi⋅1][xi⋅⊤1]∇i0=[−(ϕ(ηi1)−ϕ(ηi2)Φ(ηi1)−Φ(ηi2))xi⋅ϕ(ηi1)+ϕ(ηi2)Φ(ηi1)−Φ(ηi2)](∇i0)2=[Φ(ηi1)−Φ(ηi2)]−2[(1)(2)(2)⊤(3)]

where the matrix entries are

(1)=(−[ηi1ϕ(ηi1)−ηi2ϕ(ηi2)][Φ(ηi1)−Φ(ηi2)]−[ϕ(ηi1)−ϕ(ηi2)]2)xi⋅xi⋅⊤(2)=([ηi1ϕ(ηi1)+ηi2ϕ(ηi2)][Φ(ηi1)−Φ(ηi2)]+[ϕ2(ηi1)−ϕ2(ηi2)])xi⋅(3)=(−[ηi1ϕ(ηi1)−ηi2ϕ(ηi2)][Φ(ηi1)−Φ(ηi2)]−[ϕ(ηi1)+ϕ(ηi2)]2)

and

ηi1=t−xi⊤s2ηi2=−t−xi⊤s2S1(η)=ϕ(η)1−Φ(η)S2(η)=ϕ(η)Φ(η).

B Conditional distributions for gibbs sampling

Using the distributional assumptions for Method 2 in Section 3.2.2, the full conditional distributions are as follows. We can sample the latent performances y and the skill levels s as blocks:

yi∣s,t∼indN(sa1i−sa2i,2)1ai≤yi≤bi for i∈1,…,n,

where

ai,bi={t,∞if oi=1−t,tif oi=0−∞,−totherwises∣y∼NA((x⊤x+2σs2)−1x⊤y,(x⊤x+2σs2)−1),

and

σs2∣s∼Γ−1(aσ+A2,bσ+||s||222)

and we collapse the chain by sampling p directly conditional on o. Let |o0| be the number of ties in the data set. Then p∣o∼beta(ap+|o0|,bp+n−|o0|) and t=2(1+σs2)Φ−1(1+p2).

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Published Online: 2019-06-01

Published in Print: 2019-08-27

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Artikel in diesem Heft

https://doi.org/10.1515/jqas-2018-0103

Schlagwörter für diesen Artikel

Bradley-Terry model; game theory; generalized linear models; luck; ludology; skill