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# Martingale revoked

5 replies to this topic

### #1 bonanova

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Posted 28 July 2012 - 08:57 PM

In a game of chance a player wins his stake with a favorable outcome (W) and loses it otherwise (L).
A famous (Martingale) gambling strategy is to double the stake after every loss.
A win is thus assured, even after a (finite) string of losses.
The series of outcomes W, LW, LLW, LLLW, LLLLW, ... all win the original stake.

Nevertheless, the strategy loses, and it's a classic puzzle to show why.

Let's change the game a bit and eliminate that approach.
The player doubles his stake with every W, but must leave the game after only a single L.
Note that is the only penalty for L; the player keeps his accumulated winnings.
The player has no strategy: he plays until he must quit.

If his initial stake is \$1, and W and L are equally likely outcomes, what are his expected winnings?
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### #2 superprismatic

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Posted 29 July 2012 - 01:55 AM

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### #3 benjer3

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Posted 29 July 2012 - 09:18 AM

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### #4 plasmid

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Posted 29 July 2012 - 04:29 PM

@benjer, the player doesn't lose the accumulated winnings after losing a game. It just forces him to stop playing.

Spoiler for equation for the expected winnings

Edited by plasmid, 29 July 2012 - 10:19 PM.

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### #5 psykomakia

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Posted 31 July 2012 - 01:40 PM

@benjer, the player doesn't lose the accumulated winnings after losing a game. It just forces him to stop playing.

But he has no accumulated winnings. (Or if you prefer, the amount added to each subsequent bet matches that of the accumulated winnings, nullifying them.)
Spoiler for

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### #6 phil1882

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Posted 31 July 2012 - 04:57 PM

psykomakia, i think you keep your most recent bet after a loss as well.
in other words there's no penalty for losing other than having to stop playing.
Spoiler for my guess

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