After enjoying a weekend watching my favourite team win the FA Cup, my mind quickly turned towards next season and the possible transfer targets. Furthermore upon reading a Mourinho interview where he outlined some of his potential summer buys I was interested to understand how managers plan transfer market strategies. Therefore I’ve decided to post a blog incorporating another football analogy with an enlightening economic lesson. Some of you may be aware of two things:
1. The impending World Cup and transfer market dealings that come afterwards.
2. Game theory, a behavioural economic theory based on strategic decision making amongst rational agents. A theory told superbly by the Russell Crowe film – A Beautiful Mind on economist John Nash’s life.
Now to our transfer market problem. Let’s assume we have two teams, Arsenal and Chelsea who have a similar problem – to buy a striker(s). Both teams have a budget of £50m and two strategic choices, they can:
A. Bid for two alternative strikers, costing £17.5m each.
B. Bid for Benzema of Real Madrid for £30m.
Given their budget, each team aims to maximize their payoff(savings) given the cost of the player(s). Payoff (P) = 50-X. Where X = cost of player(s).
We also have two further assumptions based on demand & supply.
1. If one team bids for Benzema and the other for an alternative, the price of the alternative strikers increases. This happens because other club teams understand there is lower supply of quality strikers and therefore increase their price – from £17.5m to £22.5m each.
2. If both teams bid for Benzema, his value increases, simply on the assumption a higher demand for a good or service increases the price. Benzema’s value rises from £30m to £40m.
What does this mean for our model?
Now, you can also see the cumulative payoffs for each set of choices.
Using the payoff function described, above where P=50-X, each team payoff is illustrated in left(Arsenal) or right(Chelsea) dependent on the value X, e.g. if both teams buy 2 alternative strikers, Arsenal payoff = P = 50m – (17.5m x 2) = £15m. Simple enough right?
We can clearly see the dominant strategy for Arsenal and Chelsea is to bid for Benzema, that is bidding for Benzema always yields the highest individual payoff(savings) irrespective of the other team’s decision. If both teams bid for Benzema, we fall into Nash Equilibrium where neither player sees it advantageous to change their strategy unless the other player also does so.
Arsenal [A], Chelsea [A] – £30m saved
Arsenal [B], Chelsea [A] – £25m saved
Arsenal [A], Chelsea [B] – £25m saved
Arsenal [B], Chelsea [B] – £20m saved
It is also apparent that the Nash Equilibrium solution is not optimal, if both teams co-operate and bid for two separate alternatives, they will both accrue savings of £15m, a co-operative strategy that saves the highest amount for both teams £30m cumulatively.
What seems odd is from this simple economic model, we find a solution where both teams may actually do better by working together! That is, by not bidding for Benzema, Arsenal and Chelsea save more money and can buy more players.
Now I know this model relies on certain assumptions of player values and the probability of the strategy choices. It also omits variables such as player exchanges, future expected performance impact and most importantly. The assumption Arsene Wenger and Josè Mourinho will be willing to co-operate.
I still hope it provides a simple lesson on how economic methods and techniques can be used in a fun and simple way to partially solve everyday problems.