Game Theory

Game Theory Parables for Game Designers

The Dollar Auction

Up for auction is bright and shiny $1 coin.  Bidding starts at 1 cent, with increments of at least 1 cent.

The catch?  The loser pays their bid too, but receives nothing.

What should you do?

Well, if you bid 1 cent and win, you’re up 99 cents - amazing!  Your friend notices this potential profit, and decides to bid 2 cents, eying a 98 cent profit.

But now you’re in a situation where you’re out 1 cent with no return.  Might as well bid 3 cents and get that 97 cent return.  But then your friend bids 4 cents…

How long will this go for?  How long can you stand the tension between making a sizeable profit and taking a (small-but-rapidly-growing) loss?  Rational play seems to drive you to keep bidding.

There’s a point of inflection at $0.50, when the loss of losing the auction outweighs the gain of winning it (e.g. being outbid at 50 cents means either losing 50 cents, or bidding more and gaining less than 50 cents).  If players get passed this point, they’ll likely continue to the next one.

The next point of inflection is at $1.00, when both losing and winning the auction result in a loss (but the loss from losing is still much higher).  If you have to bit $1.01 to win, you’re now entering a game where you’re competing to lose the least.

It is an understatement to say that this is an interesting predicament that players have gotten themselves into, where by their own construction they are competing to lose as little money as possible.  The structure of the auction compels them further and further, with no end (save the amount of money in your pocket).

Smart players (who have read this post already!) will note a third inflection point: whether the game begins at all.  If you know where the path is almost guaranteed to head, would you take the first step?  Could you say ‘no’ to an outrageous (potential) profit though?

One “solution” to this sort of situation is collaboration.  All bidders agree to form a cabal, submit one bid, and share the rewards equally.  For example, 9 players could agree to bid 1 cent, resulting in a net profit of 99 cents.  Split 9 ways, that’s 11 cents per player.  But of course each player has the incentive to break from the cabal, bid 2 cents, and walk away with 98 cents instead of 11 cents.  But then the next player in the cabal…

As Macbeth says, “I am in blood/Stepp'd in so far that, should I wade no more,/Returning were as tedious as go o'er.”

Sam Hillier