As mentioned in the first edition of a new series of blog posts called “Awakening”, I mentioned Game Theory. I wanted to write this tangent of a post specifically outlining this interesting branch of mathematics. It is something that still interests me to this day.
I am not going to dive deep into the math. Instead this is more of a layman’s way of describing it. I can go deeper, but this is geared towards an audience where game theory is a foreign concept. So let’s begin…..
Let’s say you are on a trip with three friends. A driver, another passenger, and yourself. To determine who rides shotgun, the driver takes both hands behind their back and says the person closest to the correct number wins.
You’re up first. This is where game theory comes into play.
In this case, you’re up first but to have the best strategy to win you need to know the optimal strategy for the second guesser. The best strategy for the second guesser is to either add one or subtract one from the first guesser. For instance, if the first guess is “9”. The second guess should be “8”. The person that guessed 8 can win with any number from 1 to 8 (or 80%). Where as the first guesser can only with with a 9 or a 10 (or 20%).
If you have ever watched The Price is Right you might have a general idea on this thought process. How often do you see the highest bid at something like $500 and the last person bidding $501 because they think it is more expensive.
In the picture above, the blue contestant had the right idea. The yellow contestant, if he was to act after the blue contestant should have said “752” instead of “1120”. The Price is Right is a little different in that there is a penalty for going over the price. This is why there is also the strategy of saying “1 dollar” as on display with the red contestant.
Back to us trying to win shotgun on our road trip. Now that we know what should be the strategy of the second guesser and we want to have the best odds to win and not be taken advantage of, the correct response would be 5 or 6.
The reason is that you are guaranteed to have at least 50% of the field. Let’s say you said 5. This means that if your opponent in this game selected 4, you would win if the number is 5, 6, 7, 8, 9, or 10. That represents 60% of the possibilities.
If your opponent selected 6, you would win if the number is 5, 4, 3, 2, or 1. Or 50% of the possibilities.
The same numbers would apply if you chose 6 instead of 5. You would have a max of a 60% winning probability and a minimum of 50% winning probability, thus 55% average against another “thinking” opponent.
This is called an unexploitable strategy. You are guaranteed to not be exploited, or taken advantage of, by your opponent.
This is also considered “optimal strategy” in that it is the best play for the amount of information that we have. However, optimal strategy can change with the more knowledge we can gain on the opponent.
Let’s say that we know for a fact that the person we are bidding against for that coveted front seat on this long trip has a favorite number. This person nearly always says their favorite number in a game like this. We also know that this person’s favorite number is 3. Thus we should choose the number 5 over that of the number 6.
Did you catch something there? Yes, I deviated slightly from the strategy. If we thought the person is going to select 3, why not select 4? The reason is, as underlined above. We are unsure if this person will definitely say 3. If the number is a 4, we would tie and have to do it again so no harm. If you said 4 then you are opening the door for your opponent to say 5 and have a significant advantage.
For the math:
Saying 4 would net us a 70% winning percentage if the opponent said 3. If the opponent however decided to say 5, our winning percentage would drop to 40%. If our opponent is just as likely to say 5 as they are to say 3, then the average is a (70% + 40% = 110% / 2) 55% winning percentage.
Compare that to just saying 5. If our opponent says 3, we will win 60% of the time, tie 10% of the time and lose 30% of the time. If they deviate from their lucky number and try to guess optimally, they will choose 6. Which still give us a 50% winning percentage.
Thinking deeper, we see that (60% + 50%) / 2 = 55% winning percentage. Just like if we had said 4. Right? Yes, but throwing in the wrench of the 10% tie changes things. Not only is it important to increase our winning percentage, but we need to keep our opponent’s winning percentage as low as possible.
In scenario one, 55% winning percentage meant our opponent won 45%. In scenario two, 55% winning percentage does not equal an opponent’s winning percentage at 45%. Rather because of the possibility of a tie, the math works out to be a 40% winning percentage for our opponent.
Oh ok, here’s the math:
4 vs 3 = 30%, 4 vs 5 = 60%, or 45% average for our opponent to win.
5 vs 3 = 30%, 5 vs 6 = 50%, or 40% average for our opponent to win.
If we are 100% sure they will say 3, then yes 4 is the correct answer but with “nearly” thrown into the mix, we do not want to get exploited while trying to exploit our opponent.
You can even take the above one step further and try to guess at what likelihood our opponent will guess 3 vs 6 after our guess of 4 or 5. Since I just stated that if we are 100% sure they will say 3, then the correct play is to say 4. The question then becomes what percent does our opponent need to select 6, instead of 3, in order for our guess of 4 to be incorrect. I will let you try to figure that one out for yourself.
This is game theory in a sense. Finding the optimal play to be able to exploit our opponent. The less information we have, the less of an exploitative play we need to make. With more definitive information, the more we can institute an optimal strategy.