Lets assume everyone is rational and profit maximizing. Lets also say that when one is indifferent between bidding and not bidding one chooses to refrain from bidding to minimize administration and risk he might die in the process.
First lets consider the case a couple of you suggested: no one bids. Now that's easy, I bid $0.01, the rest of you rational people refrain from bidding and I net $9.99.
Obviously everyone not bidding can't be optimal.
Now let's consider the case where someone overbids me by bidding Y1, where $0.01<Y1<$10.00, since his EV of bidding is never higher than $10. Let Xn denote my nth bid and Yn denote his nth bid and let Xi=0 when one stops bidding, assuming there are only 2 bidders, which is without loss of generality, but I am too lazy to proof it.
Now, what is the EV of his bet? It is the infinite series Sum(n=1->oo)P(Yn>Xn+1|X1,Y1,...,Xn,Yn)(10-Yn)-P(Xn+1>Yn|X1,Y1,...,Xn-1,Yn)(Yn)
I have a hard time evaluating this equation, so I will first evaluate the simpler case in which Y2=0 (note Yi=0 implies Yk=0 for all k>=i). Then his EV becomes P(Y1>X2|X1,Y1)(10-Y1)-P(X2>Y1|X1,Y1)(Y1)=Z(10-Y1)-(1-Z)Y1=10Z-Zy1-Y1+Zy1=10Z-Y1, where Z=P(Y1>X2|X1,Y1) or in words the probability that he wins the auction with his bid. Now EV=0 => Z=Y1/10, so when he bids $5 I have to overbid him less than 50% of the time to make it a profitable bid for him. Now, my way of thinking is very weird since it is a game with complete information in which there are no mixed strategies and hence no probabilities different from 1 and 0 and I am using Yn and Xn as some sort of stochast and a normal variable at the same time, but I am just thinking out loud here. I have the sneaking suspicion that the optimal strategy is being the first to bid $0.01, but I don't know the proper way to respond to an overbid to, say $9. One could argue that a bid of $9.01 is clearly profit maximizing, but the probability of winning with such a bid is very small, considering he also has the exact same strategy available to him as me. The thing is, if a rational person would overbid, then both him and me are overbidding till infinity, but if a rational person wouldn't overbid, then none of us would be bidding at all after someone else has bid. The value of the game of the first situation is very negative, and the value of the latest 0 for all but the first one to bid something. So I think the latter should be optimal, but I don't know a solid way to proof it.