Maynard Smith's objective was a relatively simple demonstration that only costly signals could be honest.   Instead of attempting to prove a general mathematical theorem as Grafen had done, he used a particular example of communication.   By keeping this example simple (described by just a few variables), he expected that it would apply to many other instances of communication as well.

For this example, he adapted the story of Sir Philip Sidney in the battle of Zutphen.   A badly wounded soldier asked Sidney for his water bottle.   Although himself wounded, Sidney responded by offering the soldier his water because he was convinced that his wounded compatriot needed the water more than he did.

As usual, this mathematical paper (only 2 pages long) includes (1) the formulation of the problem (assumptions, variables, conditions), (2) the math, and (3) the conclusions (in terms of the original variables).   We can assume (2) is correct, and so we non-mathematicians can focus on (1) and (3).

(1) How is the problem formulated (what are the assumptions and variables)?

First, notice that the paper assumes that Sidney knows nothing about the wounded soldier's condition except what the soldier tells him.   Sidney concludes that the wounded soldier needs the water more than he does solely because the soldier says so.   This situation means that we are dealing with true communication (if Sidney could directly ascertain the soldier's condition then the problems of communication vanish).

Therefore the important questions are (1) under what conditions should the wounded soldier ask for water only if he needs it (that is, when should he tell the truth) and (2) under what conditions should Sidney give him the water only when the wounded soldier asks for it (that is, when should Sidney believe the soldier).

Also communication involves an interaction between two individuals.   Thus an individual's chances of survival or reproduction depend both on his behavior (or situation) and on his partner's behavior.   We thus should expect a table of payoffs (costs and benefits that influence survival or reproduction).

In this case the payoffs for the wounded soldier depend both on his true condition (wounded or not) and on whether or not he gets the water.   So we can make a 2 X 2 table of the four possible payoffs for the wounded soldier.   Each of the four entries in the table is the soldier's probability of surviving ...

					RESULT

				GETS		DOES NOT GET
				WATER		WATER


		NEEDY		1		0
CONDITION

		NOT NEEDY	1		V	

with 0 < V < 1

So if he gets water he always survives (probability = 1) regardless of whether he is wounded (needy) or not.   If he is wounded (needy) but does not get water, he always dies (probability = 0).   If he is not wounded and does not get water, he has a chance of surviving (0 < V < 1).   Perhaps this is plausible -- even if he is not wounded he might still die of dehydration.

What about payoffs for Sir Philip Sidney?   In this case, there are two outcomes and thus two payoffs (again probabilities of surviving) ...

	KEEPS WATER		GIVES WATER AWAY

		1			S

with 0 < S < 1

So Sidney also survives if he has the water.   If he gives it away, there is a cost (dehydration?) but he has a chance of surviving (S < 1).   The payoff to Sidney when he gives away the water is (for some reason) different than the payoff to the soldier when he is not wounded (needy) and does not get the water.   There is probably some way in which this situation might occur ... maybe officers are more (or less) likely to survive than foot soldiers!

One last assumption ... producing a signal (asking for water) has some cost for the wounded soldier.   So if he asks Sidney for water, his chance of surviving is reduced by a factor (t with 0 < t < 1).

Remember what the big question is ... can we determine the conditions for honest communication?   Under what conditions does it pay for the wounded soldier to ask for water only when he needs it?   And under what conditions does it pay for Sidney to give him the water only when asked?

(2) Is the math correct?

No mathematician has found an error in the math!

(3) Assuming the math is correct, we skip along to the conclusions.

There are two conclusions expressed in terms of the variables above ...

First, the wounded soldier asks for water only when he needs it (signal | needy), provided that ...

V + r > 1 - t + rS ... or rearranged ... V - 1 + r(1 - S) + t > 0

Second, Sidney gives him the water only when asked (respond | signal), provided that ...

1 + rV > S + r ... or rearranged ... r(V - 1) + (1 - S) > 0

There is one variable not discussed above ... r (coefficient of genealogical relatedness), which appears in Hamilton's Condition for the spread of alleles by kin selection.   Maynard Smith introduces r in order to include the possibility that Sidney and the wounded soldier are relatives (share a common ancestor).   If they are relatives, Hamilton's Condition shows that there will be some conditions in which altruism by one person or the other might apply.

So we expect less manipulation (and other forms of selfishness) between relatives than between non-relatives.   In fact, if Sidney and the wounded soldier are relatives, alleles in the wounded soldier might spread faster if he let himself die so that Sidney would have a better chance of surviving ... or vice versa ... depending on the costs and benefits and the coefficient r.

The case of non-relatives is the real test for honest communication.   To see the conditions for honesty in communication without the complication of kin selection, just set r = 0 (no genealogical relatedness of signaler and receiver).   The two conditions for honesty then become simple ...

signal | needy ... provided ... V - 1 + t > 0 ... or ... t > 1 - V

respond | signal ... provided ... 1 - S > 0 ... or ... S < 1

Since 0 < V < 1, the cost of honest signaling (t) must always be greater than zero.

Everything depends on that variable V.   If V were close to 1, then t could be close to zero.   If V = 1, then t could be infinitessimally close to zero.   Honest communication in this case would require signals with very little cost!

So this mathematical analysis confirms that honest communication occurs only when signals have a cost ... but not necessarily more than just a little cost.   This conclusion does not seem quite like Zahavi's conclusion that honesty results from extravagantly costly signals.