A game of everlasting halves

[One] night, Zeno visits you in a dream. He brings along a sack of ancient coins, which come in denominations of 1, 1/2, 1/4, 1/8, 1/16, and so on. Evidently the Eleatic currency had no smallest unit: For every coin of value 1/2[to the power]n, there is another of value 1/2[to the power]n+ 1. Zeno's bag holds exactly one coin of each denomination.

He teaches you a gambling game. First the coin of value 1 is set aside; it belongs to neither of you but will be flipped to decide the outcome of each round of play. Now the remaining coins are divided in such a way that each of you has a total initial stake of exactly 1/2. The distinctively Eleatic part of the game is the rule for setting the amount of the wager. Before each coin toss, you and Zeno each count your current holdings, and the bet is one-half of the lesser of these two amounts. Thus the first wager is 1/4. Suppose you win that toss. After the bet is paid, you have 3/4, and Zeno's fortune is reduced to 1/4; the amount of the next bet is therefore 1/8. Say Zeno wins this time; then the score stands at 5/8 for you and 3/8 for him, and the next amount at stake is 3/16. If Zeno wins again, he takes the lead, 9/16 to 7/16.

This is a game in which you can never go broke. Find out everything else you might want to know about it.