What Grumman said. It’s only a buck, so for virtually everyone, it’s logical to escalate to the point where you lose (but lose less) once you’re in a bidding war. At some point somebody will say screw it, and the other guy walks away with the buck, but that will be at a loss. Maybe it will be me who minimizes the loss (maybe not), but it will be a loss.
If it makes sense for you to bid 2 cents, then it makes sense for me to bid 3 cents (to see if you’ve realized the error of your ways ;)). Every bid up to and including $1 doesn’t lose any money, so why wouldn’t the progression continue, if that logic is true? At that point nobody wins and one guy loses.
As soon as the second person enters the bidding, it’s an overall losing proposition. So, it only makes sense to be the lone bidder, and to bid as low as possible, since logical people won’t outbid you. I think…
But that’s a key factor - As soon as the second person enters the bidding. The second bidder is not looking at the same situation the first bidder is.
Let’s posit for the moment the idea that “Never submit a bid if somebody else has bid” is an immutable rule.
With such a rule in place, the only person who can bid is the person who is making the initial bid. His options are to make a bid or pass. I think everyone would agree that if there’s only going to be a single bid, then he should make an initial bid. He can win the dollar without worrying about any counterbids.
And going further, if the initial bidder knows that nobody will counterbid, then he should make the minimal possible initial bid (one cent). Again, this is clearly the best strategy if nobody is going to make a counterbid.
So the first bidder makes an initial bid of one cent. This is the situation I described in my first post. And at this point, the logic of “Never submit a bid if somebody else has bid” fails.
Because if the second player knows that rule will be followed, he should make a two cent bid. If the rule is followed, then he will win the auction. The only way he won’t win the auction is if the first player counterbids him - and that means the first player didn’t follow the rule.
So regardless of which way the auction goes, it has to make rational sense for one of the players to make a counterbid. If the first player doesn’t make a counterbid, then the second player’s counterbid wins. And the only way the second player doesn’t win, is by the first player making a counterbid.
The conclusion is that “never submit a bid if somebody else has bid” leads to a logical paradox and therefore cannot be a rational strategy.
Optimum strategy is to try and put a feeler out to the other player for collusion (let the other player win with a low bid, see if he reciprocates for the next round). If that fails the OP’s criteria or Player 2 fails to pick up on it, strategy should shift to some form of Chronos’s strategy above. But this is really more of a game of psychology than anything.
I agree, if there is more than one auction, it ought to be possible to create a strategy by getting/giving information re: the other bidders, over time.
This seems to be assuming a finite bankroll for both players, and also probably an iterated game (with bankrolls varying in later games based on the outcome of the previous ones). But what if the game is played with some arbitrary “points” instead of dollars, with which one is allowed to go an arbitrary amount into debt? And what if it’s a one-off (or equivalently, if the scores are reset for each new round)?
Dumb Question: How is it ever a loss if you’re paying less than $1 for the dollar? (I’m assuming that the simulation ignores things like time spent, valuation of time, and on and on).
PS: My solution would be to collude with the other person. Arguably, if you’re allowed contact, and both players know the rules…if either player is allowed to bid zilch, both players stand to gain a reliable 49.5 cents, presuming someone has to bid a cent to win…or…is that cheating?
Ok, that said, assuming you have to raise your bid, and you can’t bid zero, and you want to win the dollar – the only way to know when you’ve won is to NOT be the guy who bids .99 cents, right? In other words – to bid 99 cents is to win. If its a random situation (coin toss for who bids first), I’d say always bid 99 cents, period; the other player is essentially forced to buy the dollar at a 1:1. Er, then I’m ignoring whether we’re assuming a bankroll.
Still, I’ll bet a dollar (heh) that the best bet in any given situation is one of three things: a one-cent raise (strike that, that introduces a variable – you could be the one to end up at $1) – so two possible solutions, assuming you have to raise your bid, and its random who goes first: the solution is to either bid 1, or to bid .99, depending on the bankroll factor. Uhh…I think.
Ahem. Two things:
(1) Sorry for skimming the OP and completely missing half the rules.
(2) I’d argue there’s no way to reliably predict a winning strategy without knowing a number of other variables, namely (a) the number of rounds to be played, and (b) the starting bankroll of each player.
…No?
Is the ultimate question, “How does the size of the bankroll, and/or the number of auctions, change the game?”?
Another thing – why would either player ever bid more than one dollar? If the other player bids 99 cents, and it goes to you, you bid $1.00 – the opponent has to fold, or they’re wasting money. Now, this would be a single round, and that might make sense…except neither player gains an advantage by intentionally taking a loss, right?
You might do it to punish the other player, if you believe that the other player’s strategy has unfairly deprived you of the prize. This might even be rational if you expected to play the same person again in a similar game.
No, the opponent is out 99 cents if they fold - the rule says you have to pay your highest bid, win or lose - but only one cent if they bid $1.01 and you chicken out.
IMO the best strategy is to decide beforehand what your maximum loss is, and bid no higher than that. By all means start lower if you think it will help.
The rule is not, “never outbid a counterbid.” The rule is never counterbid the first bid, because all you’ve done is ensure a losing proposition for all involved. The first player assumes everyone will follow the rule, if they’re logical. If there’s a counterbid, player two fouled it up for everyone. The jerk.
The problem is that if you know the other player will not counterbid, then it’s to your advantage to bid.
The only way to prevent rational counterbidding is to equalize the benefits of counterbidding and not counterbidding. That’s the basis of my ninety-nine cent initial bid strategy. That eliminates any incentive to counterbid.
If the first player opens with a ninety-nine cent bid, the second player has two possible options. Pass and gain nothing. Or counterbid a dollar for a prize worth a dollar, in which case his highest possible gain is once again nothing. So he has no incentive to counterbid without even considering what his opponent might do.
Yes. That’s why you should bid a penny. If someone counterbids two cents, I would immediately employ your strategy and bid $1.01. Game over, I minimized my losses and that jerk loses a penny, too. It isn’t rational to counterbid. It is rational to bid a penny in recognition of that.
No, you didn’t minimise your losses - you deliberately went from a guaranteed loss of one cent to a loss of either 1 cent or 101 cents, for no other reason than to make your opponent go from a 98 cent profit to a 2 cent loss. If you would act in this spiteful manner over a penny, what’s stopping your opponent from doing the same to you over their two penny loss?
The only rational way to bid is to bid less than X% of the prize, where the likelyhood that you are the only participant is X%. If you know that there is another participant, bidding is irrational.
Rules of the auction is that both players have to pay up whatever their bid was. So in your example, the player who bid 99cents would lose 99cents if you bid a dollar at the end. This encourages him to bid $1.01 because if he wins at that point, his losses will go from $0.99 to ($1.01-1)=.01
Of course, at that point, there’s nothing stopping you from re-upping to $1.02…
Its not the penny loss that’s the concern, its the fact that between the two of us, we both could have made the maximum average Expected Value for this game (49.5 cents each), and player 2 decided to start a dumb bidding war that will most likely end up with us both getting an average EV of 0 cents instead (or somewhere thereabouts). Player 2 needs to be browbeaten into submission, for the sake of both of us.
So if Player 2 makes any counter-offers to the 1 cent opening bid, immediately cut to the chase and bid X+99. There is now no valid reason for him to re-bid. All rational players should drop out at this point, and you’ve cut everyone’s losses. If he rebids your $1.01 offer to $1.02, counterbid with $2.01. Hopefully he’s received the message. If not, he’s completely irrational and just playing out of spite. All further strategy breaks down at this point, so its probably not worth playing any more.
As this thread clearly shows, it is not possible to strategize for a win - the best I can do is arrange for a possible win while keeping my possible loss within acceptable bounds. Making an OMGUS bid of over $1 to punish another player for the crime of bidding is certainly not rational.
The other rational option, if you know how many other players are in the game, is to generate a random number from 1 to n and bid one cent if and only if you rolled a 1; otherwise you make no bid.