Inspired by a recent thread on bid fee auction sites.
Martin Shubik invented the dollar auction. It’s a simple concept: two people are bidding for a dollar which is given to the highest bidder. It’s presumed that both people want to maximize the amount of money they have and they cannot collude together. The catch is that when the auction is completed, both people have to pay the highest amount they bid. But only the one who made the highest bid overall gets the dollar.
When it’s played, a dollar auction is a good demonstration of how sunken costs work. Once a person has started bidding they have an incentive to keep bidding.
Let’s say their current bid is thirty-three cents and the opposing bidder comes back with a thirty-four cent bid. If the first bidder drops out now, he’ll have to pay thirty-three cents and get nothing. But if the first bidder bids thirty-five cents and wins, then he’ll have a gain of sixty-five cents rather than a loss of thirty-three cents. So he makes the bid.
But his opponent faces the same situation. He has to make a choice of dropping out and losing thirty-four cents or making a thirty-six cent bid in the hope of getting a sixty-four cent profit.
The logic still applies even after the bids go over a dollar. The person who drops out will still do worse than the person who makes the final bid. Assuming they bid up one cent at a time, the person who drops out will always come out with ninety-eight cents less than the person who makes the final bid.
People playing this game tend to get caught up in it. So let’s take human emotions out of it and assume it’s a competition between two computers. They have no emotional involvement in the game. They’re just being programmed to play whatever rational strategy will achieve the maximum amount.
But what is that rational strategy?
Should both computers be programmed with the knowledge that this is a sucker’s game and they should rationally refuse to play? The dollar auction starts and neither computer makes a bid.
Or does it depend on who makes the first bid? Let’s say computer 1 is chosen at random to make the first bid. Can it make a bid and “know” that computer 2 will not respond with a higher bid?
Suppose computer 1 makes an initial bid of ninety-nine cents. Computer 2 analyzes the possibilities and calculates that if it bids nothing it will make zero profit. If it bids a dollar and computer 1 quits, it will also make a zero profit. If it bids a dollar and computer 1 responds with another bid, then regardless of which computer ends up winning they will both end up with a negative amount.
So it appears that a computer will not submit a bid if the other computer makes an initial bid of ninety-nine cents. Under that circumstance, there can be no strategy that will give it more money than doing no bidding will.
And that means that a computer can be programmed to make a ninety-nine cent bid if it has the initial bid. The other computer will be programmed to not bid against it so the first computer will make a profit of one cent.
But now comes the tricky part. Should you program a computer that has the initial bid to make a bid of one cent?
The question comes down to what the response would be from the other computer. What is the rational thing to program a computer to do if it has the second bid and the opposing computer made an initial bid of one cent? Should a computer be programmed to never bid unless it gets the initial bid? If so, then the best program would be for a computer to bid one cent if it gets the initial bid.
But now consider this. Assume as a hypothetical, that the two computers have arrived at a situation where each has submitted bids. Both computer would analyze the facts I described above for human players and realize that further bidding will not improve their profit but just sink them further into the hole. So a computer should be programmed that if it is in a situation where both computers have made a bid, it should respond by not making a bid.
Now go back a step. If both computers are programmed to stop bidding once each computer has made a bid, what is the rational way to program computer 2 when it has the second bid and computer 1 made an initial bid of one cent? In this situation computer 2 should bid two cents. Because then computer 1 will be presented with the situation that both computers have made bids so it should stop bidding. Computer 2 therefore makes a profit of ninety-eight cents.
So it appears that a computer should not make an initial bid of one cent.
This OP is getting overly long so I’ll stop now. Hopefully someone else will wish to pick up the analysis from this point.