Football – substitution

A spanish banker has made some interesting observations about a football coach’s substitution choice.

The coach can make a last substitution. He can substitute an attacker for a defender or vice-versa. With more attackers the team  more likely to score but also more likely to be scored against. Substituting a defender makes the final score less uncertain. Hence there is some link with Ellsberg’s paradox. What should the coach do? How should he decide?



A classic solution would be to estimate the probability of getting through the round, depending on the choice made. But is this right?


Pause for thought …


As the above banker observes, a ‘dilemma’ arises in something like the 2012’s last round of group C matches where the probabilities depend, reflexively, on the decisions of each other. He gives the details in terms of game theory. But what is the general approach?



The  classic approach is to set up a game between the coaches. One gets a payoff matrix from which the ‘maximin’ strategy can be determined? Is this the best approach?



If you are in doubt, then that is ‘radical uncertainty’. If not, then consider the alternative in the article: perhaps you should have been in doubt. The implications, as described in the article, have a wider importance, and not just for Spanish bankers.

See Also

Other Puzzles, and my notes on uncertainty.

Dave Marsay 

What is the Public Understanding of Risk?

What is the Public Understanding of Risk?
Risky Business: Risk and Reward Assessment in Business Decision Making
D. Simmons FIMA , MD Analytics, Willes RE

Science in Parliament, Spring 2012, Reprinted in the IMA’s Mathematics Today, Vol. 48 No. 3 June 2012

This says very little about the public understanding of risk, and is more about the understanding within insurance and reinsurance companies. It discusses the potential use of probability in legal cases, and says:

There is no reason why such [probabilistic / statistical ] tools should not be used in government.

This contrasts oddly with an article in the previous issue:

T. Johnson, Heralding a New Era in Financial Mathematics, April 2012 

This starts by referring to Keynes and goes on:

The Bank of England believes that recent developments in financial mathematics have focused on microeconomic issues, such as pricing derivatives. Their concern is whether there is the mathematics to support macroeconomic risk analysis; how the whole system works. While probability theory has an important role to play in addressing these questions, other mathematical disciplines, not usually associated with finance, could prove useful. For example, the Bank’s interest in complexity in networks and dynamical systems has been well documented.

… As well as the Bank of England’s interest in models of market failure and systemic risk, more esoteric topics such as non-ergodic dynamical systems and models of learning in markets would be interesting. Topics associated with mainstream financial mathematics could include control in the presence of liquidity constraints, Knightian uncertainty and behavioural issues and credit modelling.

Thus, there seems to be at least one area where Keynes’ notion that uncertainty cannot always be represented by a single number, probability, is still relevant. Simmons’ contention inevitably lies outside the proper scope of mathematics, and is contentious.

Simmons does say:

All assumptions behind a decision can be seen, discussed, challenged and stressed.

This is a common claim of Bayesians and other probabilists, and has great merit, particularly if one is comparing it with a status quo of relying on gut-feel. But the decision to use a probabilistic approach is not unimportant and we should consider, as Keynes does, the implicit assumptions behind it.

There are actually many different axiomatizations of probability. They all assume that the system under consideration is in some sense regular, and that one is concerned with averages. These conditions seem to apply to insurance and re-insurance, but not always to legal matters or government policy.

My own involvement in reinsurance was in the government’s covering of the market’s failure to cope with the non-stochastic risk presented by terrorism. If it were true that government could address risk in the same way as the reinsurers, what would the point of government cover be? Similarly, in finance, what is the regulatory role of governmental institutions if the probabilistic view of risk is correct? My career has largely been spent in explaining to decision-makers why the people who ultimately carry the risk have to take a different approach to limited liability companies, who can treat risk as if it were a gamble. (I tend to find the tools of Keynes, Turing and Good appropriate to ‘wider risk’.)

Hopefully the IMA president’s up-coming address will enlighten us all.

See also

Other debates, my blog, bibliography.


Dave Marsay