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June 15, 2012

Experiments: Game Theory

Follow-up to To ensure that you are honest… from Midgley, Christopher - Pointless twaddle and meaningless diatribes

Realised I could model the previous two cases of product purchase as games. So let’s do that.

Game 1:
  1. You make a bid.
  2. A price is randomly generated.
  3. If the price is below your bid, you pay the price and get the item; otherwise you pay nothing (and don’t get the item).
Model as a game:
  • Price to play the game is X
  • You consider the product worth Y
  • We assume the RNG creates numbers between 1 and 100 inclusive.
  • X% of the time, you make Y+i, where i is the number generated minus 1.
  • (100-X)% of the time, you make X.
  • Your expected reward is one hundredth of (100-X)X+\sum_{i=0}^{X-1}{Y+i}=-\frac{X^2}{2}+XY+\frac{199X}{2}.
  • For payoff we subtract the bid.

At 0, we make nothing. At 100, we make 49.5+Y-X. Our maximum here occurs at X=Y+1/2 which, as we’re limited to integer solutions, still returns the same result.

Game 2:
  1. You make a bid.
  2. A price is randomly generated.
  3. If the price is below your bid, you pay your bid and get the item; otherwise you pay nothing (and don’t get the item).
Model as a game:
  • Price to play the game is X
  • You consider the product worth Y
  • We assume the RNG creates numbers between 1 and 100 inclusive.
  • X% of the time, you make Y.
  • (100-X)% of the time, you make X.
  • Your expected reward is one hundredth of XY+(100-X)X=100X+XY-X^2.
  • For payoff we subtract the bid.

At 0, we make nothing. At 100, we make Y-X. At Y=X, we make X, which is our best.

So in conclusion we have that both have the same maxima, but the former has a higher payoff, which I suppose you’d expect.


June 14, 2012

To ensure that you are honest…

Got back into attending experiments given a brief lull in exams. One psychology, one economics. Tried to do another psychology but I’d done a similar one so that was a no-go.

The psychology one was on coffee and mountains. Mountains are, unfortunately, not my area of expertise. We were asked the height of Mount Everest and questions such as “how high would a mountain have to be to be in the top 10% of mountains?”. Perhaps it was only the 100 tallest mountains, because there are a lot of mountains.

Attached was the following statement:
“To ensure honesty in this, you will be paid according to how accurate you are.”

Precisely, it was something like 30p for 30% correct, 10p for 10% correct. But entering with no information at all (I didn’t know how tall Everest was, so I didn’t even have an upper bound – or a lower one), I decided it was to my advantage to be dishonest: I picked a number (3000m) and entered it for all the questions, thinking that it was at least somewhat likely to show up somewhere. It didn’t, and I didn’t get anything, but I’m not sure how paying for accuracy ensures you put down what you honestly think :P.

The coffee one also had a variant on the “how much would you pay for product X thing”. It normally goes like this:
  1. You make a bid.
  2. A price is randomly generated.
  3. If the price is below your bid, you pay it and get the item; otherwise you pay nothing (and don’t get the item).
    The variant was that you pay your bid instead of the randomly generated price. The former seems to encourage “what is the maximum you’d be comfortable with paying for X”, the latter “what is the average you’d like to pay for X”, with the psychological annoyance of having the random price much lower than what you bid. I wonder whether a series of ‘games’ like that would show any difference in bidding…

The second was an economics one, which are always nice, and can often pay quite well. This one paid you depending on how good you were at simple arithmetic and IQ tests, so rather convenient. In doing the IQ tests, though (Raven but with 30 seconds per question), I realised how much of an advantage it was to have done them before (I had not), as while staring at them I had no prior methods with which to work from, and glanced blindly around. Columns? Rows? Patterns? I did not know. The first question (and they appeared to be in order of difficulty) took me about 10 seconds to figure out that the pattern went down the columns, and about the remainder of the time to pick the correct answer out of the similar boxes. It was fairly fun, though.


June 21, 2011

Individual Decision Making Experiment

Participated in an experiment today. The aim was to guess what order someone else would put a series of three objects in. I thought it was similar to the game that I called “sheeple” but apparently that’s a different game now, and I can find no trace of the meaning I thought it had.

You got paid according to six (or eight) scenarios that you answered the questions on:
1. Order: 1st 2nd 3rd. roll a d3. Compare your Xth choice with someone else’s Xth choice: if they match you get £20, if they don’t you get nothing.
2. Weight: put an amount of money on each of the three choices. If a randomly chosen person chose A first, you get the amount you put on A.
3. You have a ticket that pays £20 if a random person chose A (or B) as their top choice. How much money would you sell such a ticket for?
4. 10 random people are chosen. Exactly how many do you think put each choice first? £20 if you are exactly right, else nothing.
5. Assign probabilities to the choices where the probability is the chance you think someone else would put that choice top. Roll 1d3 and spin a coin, then:
a. If heads, the choice corresponding to the number is compared with someone else’s top choice. If they match, you get £20.
b. If tails, the corresponding % chance of winning £20, else nothing.
6. You have a ticket that pays £20 if a random person chose A (or B) as their top choice. For what % of winning £20 would you exchange your ticket for?

There were 67 questions on 4 different sets of three: (cat, dog, rabbit), (red, green, blue), (bus, car, bicycle), (ruby, diamond, emerald). The words were accompanied with pictures that severely affected my order choice.

The payment was done by drawing a question number out of a bag and being paid according to that scenario. Assuming there were all 67 numbers in the bag, you are much more likely to be paid according to scenarios 3 or 6, because there were more of those questions that the others, which occurred once per set.

The first is a straight 1/3 chance (if everything is random) affected by your prediction of the opponent, so you get the best result by putting what you think.

The second lets you put a cap on the minimum amount of money you are willing to accept, and is generally the kindest of the payment methods: you get something with 100% certainty, and you’re best off saying what you think.

Three is also nice: the standard “draw less than you’d accept: get the bet; draw more: get that” deal. This one didn’t (say it had) half of the numbers negative, either, which was always an annoying part with a 50% chance (effectively) of just taking the bet. I was very risk-averse here.

Four was the cruelest: if random, a mere 1/10 chance of being correct. Even if you psychoanalysed correctly random chance would put you off a large number of times. This was much worse than any of the other options.

Five, from my understanding, will reward you equally well if you say the opposite of what you think, or indeed place a random number on everything. I would have thought “Choose a random person’s top choice. You have the percentage you assigned to that choice chance of winning £20, else nothing.” was a better scenario.

Six was like three, but I personally found I was significantly more risk-seeking – I’d be much more gutted if I threw away a winning bet for a sub-50% chance of winning, for example.


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