On Pennies From Heaven – student

student from thus spake a.k.

Recall that the Baron and Sir R-----'s most recent wager first had the Baron place three coins upon the table, choosing either heads or tails for each in turn, after which Sir R----- would follow suit. They then set to tossing coins until a run of three matched the Baron's or Sir R-----'s coins from left to right, with the Baron having three coins from Sir R----- if his made a match and Sir R----- having two from the Baron if his did.

When the Baron described the manner of play to me I immediately pointed out to him that it was Penney-Ante, which I recognised because one of my fellow students had recently employed it to enjoy a night at the tavern entirely at the expense of the rest of us! He was able to do so because it's an example of an intransitive wager in which the second player can always contrive to make a choice that will best the first player's.

Pennies From Heaven – baron m.

baron m. from thus spake a.k.

Sir R-----, my good friend! Come shake the snow from your boots and join me by the hearth for a draught of warming spirits!

And will you also join me in a wager whilst you let the fire chase the chill from your bones?

Fine fellow! Stout fellow!

I have in mind a game that reminds me of my raid upon the vault of Heaven, which I mounted in order to make amends to the Empress for my failure to snatch the Amulet of Yendor from the inner circle of Hell.

On Divisions – student

student from thus spake a.k.

The Baron's game most recent game consisted of a series of some six wagers upon the toss of an unfair coin that turned up one side nine times out of twenty and the other eleven times out of twenty at a cost of one fifth part of a coin. Sir R----- was to wager three coins from his purse upon the outcome of each toss, freely divided between heads and tails, and was to return to it twice the value he wagered correctly.

Clearly, our first task in reckoning the fairness of this game is to figure Sir R-----'s optimal strategy for placing his coins. To do this we shall need to know his expected winnings in any given round for any given placement of his coins.