Archive for game theory

asymmetric information

Posted in Kids, R with tags , , , , , on November 4, 2020 by xi'an

The Riddler of 16 October had the following puzzle:

Take a real number θ uniformly distributed over (0,100). Among three players, the winner is whoever guessed the closest price without going over θ. In the event all guesses exceeded θ, the contestant with the lowest (and therefore closest) guess is declared the winner. The second player knows the first player’s guess and the third player knows both other guesses. What is the optimal guess for the first player, assuming all players maximise their probability of winning?

Looking at the optimal solution z for the third player leads to six possible choices, depending on the connection between the other guesses, x and y. Which translates in the R code

topz=function(x,y){
  if((2*y>=x)&(y>=1-x))  z=y-.001
  if(max(4*y,1+y)<=2*x)  z=y+.001
  if((2*x<=1+y)&(x<=1-y))z=x+.001
  z}
  
third=function(x,y) ifelse(y<x,topz(x,y),topz(y,x))

For there, the optimal choice y for the second player follows and happens on a boundary of one of the six regions, which itself returns that the optimal choice for the first player is x=2/3, leading to equal chances of winning (although there is some uncertainty on the boundaries). It is thus feasible to beat the asymmetric information. The picture above was my attempt at representing the probabilities of gain for all three players, some of the six regions being clearly visible, with first axis being x and second being y [and z is one of x⁻,x⁺,y⁻,y⁺]. The R code is too pedestrian to be reproduced!

the biggest bluff [not a book review]

Posted in Books with tags , , , , , , , , , , , on August 14, 2020 by xi'an

It came as a surprise to me that the book reviewed in the book review section of Nature of 25 June was a personal account of a professional poker player, The Biggest Bluff by Maria Konnikova.  (Surprise enough to write a blog entry!) As I see very little scientific impetus in studying the psychology of poker players and the associated decision making. Obviously, this is not a book review, but a review of the book review. (Although the NYT published a rather extensive extract of the book, from which I cannot detect anything deep from a game-theory viewpoint. Apart from the maybe-not-so-deep message that psychology matters a lot in poker…) Which does not bring much incentive for those uninterested (or worse) in money games like poker. Even when “a heap of Bayesian model-building [is] thrown in”, as the review mixes randomness and luck, while seeing the book as teaching the reader “how to play the game of life”, a type of self-improvement vending line one hardly expects to read in a scientific journal. (But again I have never understood the point in playing poker…)

Covid’s game-of-life

Posted in Books, pictures, Travel, University life with tags , , , , , , , on May 7, 2020 by xi'an

A colleague from Paris Dauphine, Miquel Oliu-Barton made a proposal in Le Monde for an easing of quarantine that sounds somehow like Conway’s game of life. The notion is to define a partition of the country into geographical zones with green versus red labels, representing the absence versus presence of contagious individuals. With weekly updates depending on the observed cases or the absence thereof. While this is a nice construct that can be processed as a game theory problem, I am not so sure that it fits the specific dynamics of the coronavirus, which is not immediately detected while active, hence inducing a loss of efficiency in returning quickly enough to a red status. Not mentioning the unreliability and unavailability of tests at this scale. Or an open society (as opposed to China or Vietnam) where (a) people will resent local lockdown more than they do with global lockdown and (b) mostly operate outside the box for work or family interactions.

Le Monde puzzle [#1115]

Posted in Kids, R with tags , , , , , on October 28, 2019 by xi'an

A two-person game as Le weekly Monde current mathematical puzzle:

Two players Amaruq and Atiqtalik are in a game with n tokens where Amaruq chooses a number 1<A<10 and then Atiqtalik chooses a different 1<B<10, and then each in her turn takes either 1, A or B tokens out of the pile.The player taking the last token wins. If n=150, who between Amaruq and Atiqtalik win if both are acting in an optimal manner? Same question for n=210.

The run of a brute force R code like

B=rep(-1,200);B[1:9]=1
for (i in 10:200){
    v=matrix(-2,9,9)
    for (b in 2:9){
       for (a in (2:9)[-b+1])
       for (d in c(1,a,b)){
        e=i-d-c(1,a,b)
        if (max(!e)){v[a,b]=max(-1,v[a,b])}else{
         if (max(e)>0) v[a,b]=max(v[a,b],min(B[e[which(e>0)]]))}}
     B[i]=max(B[i],min(v[v[,b]>-2,b]))}

always produces 1’s in B, which means the first player wins no matter… I thus found out (from the published solution) that my interpretation of the game rules were wrong. The values A and B are fixed once for all and each player only has the choice between withdrawing 1, A, and B on her turn. With the following code showing that Amaruq looses both times.

B=rep(1,210)
for(b in(2:9))
 for(a in(2:9)[-b+1])
  for(i in(2:210)){
   be=-2
   for(d in c(1,a,b)){
    if (d==i){best=1}else{
      e=i-d-c(1,a,b)
      if (max(!e)){be=max(-1,be)}else{
       if (max(e)>0)be=max(be,min(B[e[which(e>0)]]))}}}
   B[i]=be}

Le Monde puzzle [#1085]

Posted in Books, Kids, R with tags , , , , , on February 18, 2019 by xi'an

A new Le Monde mathematical puzzle in the digit category:

Given 13 arbitrary relative integers chosen by Bo, Abigail can select any subset of them to be drifted by plus or minus one by Bo, repeatedly until Abigail reaches the largest possible number N of multiples of 5. What is the minimal possible value of N under the assumption that Bo tries to minimise it?

I got stuck on that one, as building a recursive functiion led me nowhere: the potential for infinite loop (add one, subtract one, add one, …) rather than memory issues forced me into a finite horizon for the R function, which then did not return anything substantial in a manageable time. Over the week and the swimming sessions, I thought of simplifying the steps, like (a) work modulo 5, (b) bias moves towards 1 or 4, away from 2 and 3, by keeping only one entry in 2 and 3, and all but one at 1 and 4, but could only produce five 0’s upon a sequence of attempts… With the intuition that only 3 entries should remain in the end, which was comforted by Le Monde solution the week after.