More Coin Tossing

You have the opportunity to play a game with a biased coin, where $P(H)=\frac{1}{2}+b$ and $P(T)=\frac{1}{2}-b$. In each round of the game, you toss the coin and win or lose one dollar. The game is played as follows. 

On the first toss: receive a dollar regardless of the outcome. 
On all other tosses: if you won (lost) a dollar on the prior toss, you win (lose) a dollar if the current toss comes out heads and lose (win) a dollar if it comes out tails.

What would you pay to play this game with a coin of bias $b$ for exactly $n$ rounds? Why?

Prediction Markets

Prediction markets trade in contracts that pay off according to whether some future event occurs. If the event occurs the contract pays off 1 unit, otherwise the contract becomes worthless. As a result, the price of a contract can be interpreted as the market's estimate of the probability that the future event will occur. Assume frictionless markets: one can buy (risking $r$ units to win $1-r$ units) and sell (risking $1-r$ units to win $r$ units) contracts at no cost.

QUESTIONS

1.) Individuals $A$ and $B$ think the probability that a future event will occur is $p$ and $q$ respectively. $A$ and $B$ have risk capital $X$ and $Y$. $A$ and $B$ invest so as to maximize the expected log of their capital. What price $r$ for the contract will assure that the sum of the demand for the contract from $A$ and $B$ is 0?

2.) A future event has $n$ possible outcomes each with associated contract $c_{i}$ which will pay off 1 unit if the $i$th possibility occurs while the other $n-1$ contracts will become worthless. Each contract has a market price $m_{i}$ and the market is efficient: $\sum_{i}^{n} m_{i}=1$.

You believe the true probability of the $i$th possibility occurring is $q_{i}$ for $i=1,...,n$ (where the $q_{i}$ also sum to 1). You have $X$ units to invest and you wish to do this in such a way that the expected value of the logarithm of your capital is maximized. How many of each of the $n$ contracts should you buy?

Poker Dice

In Poker Dice, the aim is to build the best possible poker hand, where straights and flushes don't count. The first person to roll has up to three throws of the dice and after each throw can put aside any dice they wish to use for their hand; also, they can stop after the first of second roll if they wish. After the first person establishes a number of rolls, each successive player may only throw the dice at most that many times. The player with the highest ranking hand after everyone has had a turn wins the game. Is there any way to gain an edge?

1-4-24 (Midnight)

How to play

Midnight is played with six dice where the objective is to get the highest score. Someone starts by rolling six dice. On each roll, you are required to keep at least one of your dice, and you re-roll the remaining ones. You do this until all six dice have been kept. So at most you will roll six times.  For example, you can choose to keep anywhere from one to six dice on the first roll. If you kept two of them, then you'd re-roll the remaining four dice, and you can choose to keep one to four of them. And, so on...

The sum of the six selected dice becomes your score, with one caviat: you must qualify the hand by rolling a one and a four. Therefore a maximum score is twenty-four by rolling 1-4-6-6-6-6, and a minimum score is zero by not qualifying.

When beginning, the order is determined randomly. After a round has been played, the lowest score goes first and the highest score goes last. Clearly, going last has the advantage due to the ability to know when to simply just stop trying for a  higher score while taking on the risk of getting a lower one. Say the first person scores an 8. And, as the 2nd player, your very first roll is 1-4-6-5-1, you can simply stop right there and claim a win with a score of 12.

For a three, four, or more player game, only a single distinct winner can win the pot. So, say in a four player game, two people score 18, one guy scores 15, and another scores a 13. Since there is no distinct winner due to two best scores tying, there is no winner this round. All players ante up again, building the pot. All players are still in the game, and therefore, are eligible to win. You play until there is a single distinct winner, who scoops.

Is there an optimal strategy for this game? If so, what is it?

Rock Paper Scissors

The best game ever? If you don't know RPS, I feel bad for you. What is the optimal way to play? Is there a way to get an advantage? I think so...

Optimal Proportional Play

Question

You are at a craps table next to a gambler looking for some action. Both the table and the die are fair. She offers you the following "even money" proposition: starting with the next roll (where the sum of the die's faces are tallied) bet whether two consecutive 7s or a 12 is rolled first. You can pick either side and the size of the bet.


What side of the proposition would you take and how would you wager?

Optimization

Question

You are bidding for an asset that you can immediately sell for 1.5 times what you pay for it but don't know exactly what the seller believes it is worth. You are told the seller's value can be accurately modeled as a random variable uniformly distributed between 375 and 1000. You can only make one bid, and the seller will only trade if your bid is higher than what they believe the asset is worth. What is your bid?