Thursday, 22 January 2015

Show More, Take All

There is a deservedly-obscure bravado-driven pub game called 'Show More, Take All', in which the two players compare the contents of their wallets, and the person with the most money takes all of the other person's cash.  Like a lot of asymmetric-information games, it has in interesting solution. Abstractly, the game is modelled as follows: each player has an amount of money known only to them, and a choice either to 'play' or 'not play'.  Only if both players choose 'play' is the money transferred, from the person with the least to the person with the most.  

The game is potentially-terrifying.  One of Aleph Insights' correspondents writes that he was once offered a game of SMTA by a friend-of-a-friend in a pub.  Coincidentally, he'd just drawn out £700 as he was travelling abroad and was about to get foreign exchange.  He thought he therefore stood a very good chance of winning and accepted.  His interlocutor, it turns out, was a builder who had just finished a job and was carrying £1500 in his wallet.  Only a last-minute attack of conscience on the builder's part prevented the game from going ahead.

You might start by thinking that a good strategy would be to 'play' if you have a large amount of money, and 'not play' if you have a small amount of money.  This kind of strategy can be specified by a single number (call it 'x') which is the cut-off below which you don't play.  

The trouble with a strategy like this is that, if one player is playing it, the other person then has an incentive not to play if their amount of money is close to 'x'.  The optimal cut-off against someone playing with a strategy 'x' is going to be something higher than x - call it 'x1'.  In turn, of course, the strategy that beats strategy x1 will be a cut-off even higher than x1 - call it 'x2'.  And so on.

The upshot is that there is no cut-off that can be a best strategy for both players.  The equilibrium is for both players to 'not play', regardless of how much money they have.  Intuitively, the only person you would want to play against is someone that didn't want to play themselves.  In the words of the WOPR from 'WarGames', the only way to win is not to play. 

This perhaps-surprising solution raises a puzzle as to why superficially-similar real-world situations can occur.  One obvious analogue is armed conflict: people should only enter into conflicts that they know they're going to win; why, then, are there always two willing parties?  They can't both be right. What are the features of armed conflict that make it differ relevantly from 'Show More, Take All' that mean it is still a viable course of action?

(Photo: Cecil Beaton)

"Never, never, never believe any war will be smooth and easy, or that anyone who embarks on the strange voyage can measure the tides and hurricanes he will encounter. The statesman who yields to war fever must realise that once the signal is given, he is no longer the master of policy but the slave of unforeseeable and uncontrollable events. Antiquated War Offices, weak, incompetent, or arrogant Commanders, untrustworthy allies, hostile neutrals, malignant Fortune, ugly surprises, awful miscalculations — all take their seats at the Council Board on the morrow of a declaration of war. Always remember, however sure you are that you could easily win, that there would not be a war if the other man did not think he also had a chance." - Winston Churchill


Anonymous said...

The expected pay-off for declining an offer of SMTA is zero (with zero variance too). What is the pay-off for declining an offer of war? For example, one made by a hostile power invading the territory you rule.

Nick Hare said...

This is a good point. But isn't the payoff from declining the offer still going to dominate the payoff from accepting and losing?

Nick Hare said...
This comment has been removed by the author.
Unknown said...

I'd like to know whether there some long-term strategic benefit in always playing Show More Take All with precisely nothing in your wallet. This way, you always lose, but lose nothing. Which means that you can freely build up data on the opposition, which may have future cash value.