In a Rock Paper Scissors game which your opponent is not allowed to play Rock, the loser of each game needs to give the winner $1, how much would you spend to play it, and why?
All of my wealth which is $18,753,704.
Why can't I just spend all my money and play scissors every time? Doesn't that guarantee that I win or tie every game?
Let say you can’t just play scissors all the time because your opponent will learn and response with playing scissors only, then the fair value of the game is 1/4.
I will spend nothing as I will never lose
If you never lose you will not get a lot of money. The only way to not lose is to pull scissors 100% of the time, so your opponent will end up doing the same. To make money, you should gamble from times to times and throw a rock, and your opponent will throw a paper from times to times too. It's a small risk, but you will lose a few times.
Pointless. Because your opponent and you will all ways both throw scissors. You will make 0 money.
That's not true, you can (and should) throw rocks fairly often. This has 2 advantages: 1. Get easy wins because your opponent will most likely pull scissors most of the time. 2. Force your opponent to pull paper, which will give you other wins when you switch back to scissor. Yeah, you will lose a few times, but the return on investment will be even greater. It's a no brainer: the risk is so low and the returns potential so high.
If they catch on and try to play scissors you can still gamble and play rock.
Assuming they pick Paper or Scissors uniformly at random: I would play scissors. I would win 50% of the time (against paper), and draw 50% of the time (against scissors). Let’s say you spend $x per play. We want our net gains to be positive. So, we have: 0.5(-x+1) + 0.5(-x) > 0 -x + 0.5 > 0 x < 0.5 So, I would pay less than $0.5 per game to play.
Do you both lose your payment in a tie?
1/3 of a dollar. You want to find the Nash equilibrium, which is that you play scissors with probability 2/3 and rock with probability 1/3, and they play 2/3 scissors and 1/3 paper. The expected payoff for you is 1/3. You can verify that if either player changes their strategy, the expected outcome doesn’t improve. What position were you interviewing for?
This is dope.
Unlimited dollars. You have a zero risk option (scissors) that at worst results in a draw, at best wins a dollar. That becomes the default option and results in a risk free rate of return. Some small percentage of turns can be rock, which will force the opponent to either concede or attempt to match based on probability guessing. With randomization this skews both options to a positive return.
But you are still paying per play (right?)