To bid or not to bid? – Game Theory for lawyers I

*Note – I wrote this a month or so ago and didn’t post it as I didn’t think the message was that clear. Given the announcement from Grayling today (1st July) that choice is back in and the Law Society’s coincidental(?) announcement of an alternative proposal, I thought I might as well resurrect it as the message is still a valid one.*


Before I got disastrously sidetracked into a career in the law, I did a degree in maths (looking back now, think of all the things that I could have done with it … ). One area that I took a couple of courses in was called ‘game theory’. It was made popular a couple of years ago in the film ‘A Beautiful Mind’.

It’s pretty interesting – it’s a way of bringing a mathematical approach to the analysis of decision making. Most people have heard of the Prisoner’s Dilemma (if you haven’t, then start here), but I’m going to have a look at a slightly different game – the Volunteer’s Dilemma. It’s basically a variant on a multi-player Prisoner’s Dilemma.

Rather than just re-inventing the wheel, I’ll quote from Wikipedia – “One example is a scenario in which the electricity has gone out for an entire neighborhood. All inhabitants know that the electricity company will fix the problem as long as at least one person calls to notify them, at some cost. If no one volunteers, the worst possible outcome is obtained for all participants. If any one person elects to volunteer, the rest benefit by not doing so”.

What’s this got to do with law?

It’s to do with BVT and the current consultation. One way of defeating it that has been mooted is for no-one to bid for a contract (see the story in the Gazette from 6th May 2013 that only 3 of the ‘top 25’ will). This is a tempting idea but will it work? Currently there are 1,600 firms. Some of these won’t bid for a contract for other reasons – they’ve had enough. Almost all will be too big or too small, so would have to merge or split. Additionally, there are some potential ‘new entrants’ – Stobarts, other non-law companies, and barristers chambers.

Say then there will be more than 1,600 entities bidding for 400 contracts. Looking at London West & Central where there are 38 contracts available, there are probably about 200 entities who would be thinking about bidding for a contract. Trying to get people not to bid is a classic Volunteer’s Dilemma.

You’re sitting there working out what to do, you know that the decision as to whether to bid or not has to be made. And it has to made in ignorance of what everyone else is doing. People have been saying that they won’t be bidding. But, how to put this, do you trust everyone else?

If no-one bids, great. We all get the benefit of defeating BVT. If two or three people bid, then it’ll probably be ok, that’s not enough for the MoJ to carry on. The problem comes if it turns out that other people have been bidding, just in case.

Maybe they weren’t planning on it. Maybe they don’t want to, it’s just that if you don’t bid and BVT comes in, you are well and truly stuffed. So, maybe they were forced to. Maybe they thought to themselves that if it does come in, at least if they have a contract then they will do the job properly, unlike Flywheel, Shyster & Flywheel down the road (or MultiNational PLC on the trading estate), so they really owe it to the public to bid.

You know what the other people are thinking, so it’s not as easy as saying that you won’t and trust that others won’t too. If you do bid, then it’s likely that there will be well under or well over 38 bidders in which case, hey, it’s not happening anyway (or it would be anyway without your bid). Nothing lost. After all, if there are 38 contracts, you’re not likely to be the one person that tipped it over the edge – being the 38th bidder.

So, on that basis, you might as well bid? Of course, if everyone thinks like that …

And there you go, you’re in the middle of a Prisoner’s Dilemma.

What happens in real life?

In 1984 Science 84 magazine ran a contest related to this. The rules were simple. Everyone writes in to the magazine asking for $20 or $100. Everyone gets what they asked for, provided (and here’s the catch) no more than 20% of people ask for $100. If that happens, you all get nothing.

If everyone asks for $20, then everyone goes home happy. But, of course, if there are loads of other people entering, all asking for $20 then great – you ask for $100 and get that and everyone goes home happy.

What happened? There were 33,511 entries. Of that, 11,758 went for $100 (35%) and as a consequence, they all went home empty handed.

This is similar to our situation with BVT, except the difference between the ‘payoffs’ is not $20 and $100, but a lot more extreme.

The problem with a Prisoner’s Dilemma is that the optimum strategy is often one that is not the best for you individually. In other words, logic sometimes dictates that, to preserve your own position, you don’t go for the best option for the fear of being shafted. Or, as Italo Calvino put it more eloquently, “you know that the best you can expect is to avoid the worst.

Can I have a bit more maths please?

Of course. The way of analysing these ‘games’ is through the use of a ‘payoff table’. Here, I’ve used C to stand for ‘co-operating’ – ie, going along with the idea of not putting a bid in, and D to stand for ‘defecting’ – putting in a bid when others are saying that you shouldn’t.

                       .No Bid                        Bid

No Bid 10, 10 2, 8
Bid 8, 2 4, 4

Your action is on the left hand side, and that of the other potential bidders on the top. For convenience, and the fact that this is not a maths blog, I’ve treated it as a two player game of you versus the world.

What do the numbers mean? I’ve put those in as a demonstration. They are arbitrary figures, but are supposed to be reflective of the outcomes that I want. So, the top left is ‘co-operate, co-operate’, In other words, this is the situation when I don’t bid and neither does anyone else does, BVT gets scrapped and we all go home vaguely happy. That’s the best outcome.

The worst outcome (2) is that I don’t bid but sufficient people do so that the tendering process goes ahead, that’s the top right. The bottom left is where I put in a bid but most people don’t. Although BVT doesn’t go ahead, it may be that my proposal to bid is ‘out there’ and people shun me down the pub. The last option, everyone defects, is the bottom right. It’s bad, because BVT comes in, but at least I’m in the tent pissing out, rather than the other way round.

This is obviously a very rough approximation. Different people may have a different take on the figures (I would suggest having a stab at putting in figures for your expected salary/earnings in each of the different outcomes). It may not be ‘symmetric’, it is not always possible to reduce it to figures and, you are not dealing with a 2-player, but a multi-player game.

It could of course be said that the figures are so bad that people cannot afford to bid and so any numbers you put in are academic. Well, leaving aside the fact that we know Stobarts are planning to bid (and therefore presumably others will be), it is interesting to note that from the Gazette article that whilst all 25 firms said the proposals are ‘unworkable’, three at least will bid in any event.

So, this is a Prisoner’s Dilemma?

It’s actually, with the figures I use, a slightly different variant called a ‘stag hunt’. Whilst you might dispute the actual figures, the ranking of the four outcomes are something that you would probably agree with.

A stag hunt is actually better than a Prisoner’s Dilemma, at least in the sense that there are two Nash equilibria, one of which is the ‘co-operate, co-operate’, so to that extent it is a lot easier to ‘play’ than a Prisoner’s Dilemma – there is a higher incentive to co-operate.

If you put the numbers in and assess the probability of the other player (in fact, groups of players) co-operating, you can work out whether you should bid. If we let P= probability that there will be sufficient bids from the other people so as to make it work (in other words, that the others will defect) then we can work out the expected values of the benefit to us.

So, E(C) = 10(1-P) + 2P and E(D) = 8(1-P) + 4P.

Tidying it up, E(C) = 10-8P and E(D) = 8-4P

Solving for P gives P=0.5 On that basis, if you assess that the Probability of other people bidding is higher than 50%, the economically rational thing to do is to defect. The exact ‘tipping point’ depends obviously on all the figures above.

Two warning notes. Firstly, in a ‘one off’ game (such as this tendering process), people are more likely to defect as there is no long term incentive. Secondly, the more people that are involved, the less societal pressure there is on one to co-operate, so the higher the chance that people will defect.

What happens next?

Well, we will have to wait and see. If the MoJ do decide to press ahead and people do bid, the above may explain why that is so.

And, if we get to the ‘auction’ stage, that is another area that game theory is ripe for.

You can read more about this in William Poundstone’s excellent book ‘Prisoner’s Dilemma


4 thoughts on “To bid or not to bid? – Game Theory for lawyers I

  1. Pingback: Save UK justice: the blogs | ilegality

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  3. Pingback: If this is a victory for the bar it certainly doesn't feel like one | Barrister BloggerBarrister Blogger

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