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A discrete random walk

Posted By: Maik Stiebler
Date: Thursday, 23 July 2009, at 10:04 p.m.

In Response To: Cube error rates (Frank Berger)

Mhh, a continuous random walk is a weird model to discuss these kinds of things, because the errors a late doubler can make are infinitely small, and tend to come in infinitely large numbers.

Incidentally I had prepared a puzzle involving a discrete random walk to post at some point in this thread:

Players A and B play a game of StaticishRace. A game consists of tossing a fair coin and changing the score by +1 or -1 respectively based on the result of the coin toss. The starting score is 0. Player A wins one point and the game ends when the score reaches 50. Player B wins one point and the game ends when the score reaches -50. Before each coin toss, Player A (only player A!, to keep things simple) is given the opportunity to double the stakes, after which Player B can either take or drop (ending the game and losing a single point). A double is allowed only once in a game.

1. What is the optimal strategy for both players?

2. What is the theoretical(=assuming optimal play from both sides) equity of the starting position?

3. Assume Player A deviates from optimal strategy by doubling if and only if the current score is +30. What is the practical equity of the starting position then?

4. How much does Player A's deviation from perfect play cost him on average per game?

Now put yourself in the position of a bot that knows the optimal strategy and observes the game, not knowing Player A's complete strategy, but noting the wrong plays that follow from the strategy.

5. a) At which points in the game does Player A, following the non-optimal strategy, blunder away theoretical equity by making a wrong play? b) How much equity does each of these wrong plays lose?

6. How often will the opportunity for Player A to make a wrong (=equity losing) play arise in a game? Compute both an average value (a) and a distribution (b).

7. Verify that the average number of blunder opportunities (6a) times the cost of a blunder (7b) equals the total cost of Player A's misguided strategy (4).

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