A lowest unique bid auction operates under the following three main rules:
As it turns out there are real sites out there that offer this type of auction, see e.g bidbudgie. Their income is usually generated by attracting lots of bidders and placing a charge for making bids. This way they are able to give away iPhones, Xboxes, Laptops and other expensive items for a tiny nominal amount and also make a profit provided enough people participate. In an article I read somewhere a gold bullion worth more than 1,000£ was sold away for only 1p because no one else bid that amount thinking it was “too obvious” (perhaps one of the best marketing gimmicks the auction company could ever wish for).
A natural question is: Are there any good strategies for participating in this auction?
On the face of it, this doesn’t look vastly different from any old lottery – you are paying a fee to choose a number on which you may or may not win.
One obvious difference is that this game has no proclaimed randomizing device such as would be found in a lottery to select between numbers, or as is the function of a deck in a card game or a die in a dice game. Instead chance is introduced exclusively via uncertainty about bids of other participants (i.e imperfect information). Therefore it seems more natural to take a game theoretic view rather than a purely probabilistic one since participants would want to take into account the strategy of others and adjust their choice accordingly. So instead of assuming a bidding distribution to model the situation we should be looking to understand what the distribution should be given some equilibrium state where everyone is reluctant to change their strategy because it would decrease their expected payoff.
Let’s make a game theoretic definition that distinguishes between two different kinds of strategies.
A pure strategy defines a specific move or action that a player will follow in every possible attainable situation in a game. Such moves may not be random, or drawn from a distribution, as in the case of mixed strategies (gametheory.net)
A pure strategy in this game is very simple since there is only one attainable situation in the game, and that is the choice of a unique fixed number .
[Alternatively one can view this as a mixed strategy choosing with probability 100%]
To simplify the discussion a bit let us make the assumption that only one bid per person is allowed so that we may assume bids are placed independently of each other. We will discuss the general case briefly at the end of this post.
Let us first consider whether there can be a “magic number” which you could always select beforehand to maximize your chances of winning the auction.
In other words we are asking whether an optimal pure strategy exists.
Suppose an optimal pure strategy did exist.
Then every other player would want to adopt the same strategy since it is optimal.
But that means they will all choose the same number and are therefore guaranteed to lose the auction.
So any strategy choosing a different number would be better, and so the original strategy couldn’t have been optimal since we have found a better strategy.
This is a contradiction.
Hence there can be no optimal pure strategy.
So having ruled out the existence of good pure strategies we turn our attention to mixed strategies. These strategies make choices from random distributions, i.e consist of some convex combination of pure strategies.
We want a strategy that somehow describes how rational participants should play this auction. One such definition is that of a Nash Equilibrium.
Informally a set of strategies is a Nash equilibrium if no player can do any better by unilaterally changing his or her strategy.
Let’s digress for a bit to discuss this concept and its potential weaknesses.
So now back to our auction.
To simplify discussion again we shall only be looking for symmetric mixed strategy Nash equilibria i.e an equilibrium where everyone chooses the same strategy in such a way that anyone deviating from it would be worse off. We shall also require that only a countable number of bids are allowed, otherwise the auction gets ridiculous.
Online the convention seems to be that the smallest increment allowed is 1 cent (or 1p if it is a UK auction).
Within most lowest unique bid auctions in the real world you are allowed to place multiple bids. Perhaps it might be interesting to discuss some potential heuristics for participating in such an auction.
On the auction site I mentioned earlier (bidbudgie.co.uk) they auction MacBook Pro Laptops and charge 3£ per bid. They claim the laptop is worth 1,889£ (according to RRP). That means they need to attract at least 630 bids to make a profit assuming 1,889£ is what they paid for the laptop (which is probably not entirely true). Looking at it’s current status on the site at the time of writing, I see there are 234 bids already from 57 unique bidders after 37% of the auction time has elapsed.
The site lets you know your rank in the auction if your bid is unique (meaning if you still have a chance at winning). The site also gives you the opportunity to buy information such as whether your bid is too high or too low in relation to the current winning bid, which costs 20p.
This means you can locate the current winning bid using binary search for under 10x(3£ + 0.2£) assuming winning bid is under 10.24£.
Looking at the tips & tricks page on the site I can see that they have a recommended strategy which is to place bids according to some arithmetic progression, say 0.13p, 0.23p, 0.33p, … , 1.13£. Then starting from your lowest unique bid you have an idea in what range the winner is currently in. You can then for example place multiple bids from 1p up to your lowest unique bid to eliminate all other players making you the current winner. Of course if you are the only person employing this strategy it may not be very expensive in comparison to the prize. However if everyone is doing this (which presumably is what the site wants) then playing this game can get very expensive and you might get carried away.
Let’s look at another crude strategy.
Suppose you are willing to invest 300£ in buying this laptop which would still save you 1,589£ if you win.
Moreover let’s say you pick all bids from 1p to 1£.
How likely is it that you will win this auction if there are 1000 bids placed on the laptop and everyone except you bids no more than once?
In a paper entitled Equilibrium strategy and population-size eﬀects in lowest unique bid auctions [Pigolotti et al (2012)] the effects of the number of participants is investigated on the symmetric Nash Equilibrium. The paper establishes that provided the number of bidders in the auction varies according to a Poisson distribution then the number of bids on each number is also Poisson in a symmetric Nash Equilibrium. When compared to real data it was found that in the region of 200 bidders this model approximates the empirical distribution fairly well and that the winning chances are evenly spread among all numbers. This means the auction behaves like a lottery for this region. For a larger number of participants they found the data begins to deviate from the model and starts following an exponential distribution because players fail to adapt to the optimal strategy and so the winning chances are highly dependent on the number of participants. On page 4 they display a very interesting log distribution contrasting the empirical winning number from theoretical.
Reading the graph it appears the winning amount should be somewhere around 80p to 1£ with 1000 participants, both from a theoretical standpoint and typically according to the empirical data (if I am interpreting it correctly). So placing a series of bids in that region for the laptop may not be such a terrible investment in an auction with a moderate number of participants where you are the most aggressive bidder.