经济代写|博弈论代写Game Theory代考|ECON2112

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经济代写|博弈论代写Game Theory代考|The Handicap Principle

The idea that differential costs of signalling can maintain honesty is known as the handicap principle. This idea was first put forward in economics by Spence (1973), who applied it to the job market. He argued that an able person paid less of a cost to obtain a higher educational qualification than a less able person, so that the qualification acted as an informative signal of ability to employers. Independently, Zahavi used the idea of differential costs to explain the preference of females for mates that have exaggerated traits such as the large antlers of male deer and the elaborate tails of male peacocks (Zahavi, 1975, 1977). Zahavi argued that such traits lowered the survival of the male that possessed them. The trait thus acted as a test of male quality: an individual with a well-developed sexually selected character is an individual who has survived a test. By choosing a male with the most developed trait the female ensures that she obtains a mate of high quality. Zahavi’s arguments for male characters as a handicap were purely verbal and many were not convinced by them until Grafen (1990) used a formal model to confirm that handicaps could work.

To illustrate the handicap principle in the clutch size model we set $b=2$. Thus all eggs survive and all males prefer their mate to lay two eggs rather than one egg. Suppose that the resident female strategy is to lay one egg when the signal is $s_1$ and lay two eggs when the signal is $s_2$. Then a low-ability male would obtain payoff 1 if he signalled $s_1$ and payoff $2-c_L$ if he signalled $s_2$. Thus if $c_L>1$ his best signal is $s_1$. Similarly, if $c_H<1$ a high-ability male does best to signal $s_2$. Thus if $c_H<1<c_L$ we have a signalling equilibrium at which males of each type are signalling optimally given the response strategy of females and females are responding optimally given the signalling strategy of males. This is also a separating equilibrium in that a female knows the ability of the male from his signal; in that sense the signal is honest. The honesty of signalling is maintained since high-ability males pay less of a cost to signal $s_2$ than low-ability males; i.e. this is an example of the handicap principle.

Since the work of Grafen (1990) a variety of models have investigated the effects of differential signalling costs and shown that these can result in a signalling equilibrium in which signals are informative even though they do not always provide perfect information (e.g. Johnstone and Grafen, 1993; Számadó, 1999; Bergstrom et al., 2002).

经济代写|博弈论代写Game Theory代考|Costs of Deviation

In our example a low-ability male is less good at provisioning young. When two eggs are laid by the female this might result in complete nest failure, even when the female helps the male to care. Alternatively the female may just desert if after having laid two eggs she discovers that the male is of low ability. We may capture these scenarios by setting $b<1$, so that if the female lays two eggs and the male is low ability then less than one offspring survives. We may also set $K=0$ for illustrative purposes. We then seek a signalling equilibrium when there are no costs to male signalling; i.e. $c_L=c_H=0$.

Assume that the female lays two eggs if and only if she receives the signal $s_2$. Then the best signal for a low-ability male is $s_1$ since $b<1$ and the best signal for a highability male is $s_2$ since $1<2$. Given this signalling strategy females are following the best response strategy. So again we have a separating signalling equilibrium.

It was originally claimed that signals had to be costly to be honest (Zahavi, 1975; Grafen, 1990). However, as Enquist (1985) and Hurd (1995) point out, and our simple example illustrates, this need not be the case. In our example, if at the signalling equilibrium a low-ability male signals $s_2$ then the male pays the cost after the female lays two eggs. This is therefore an example of what Kirmani and Rao (2000) refer to as a default-contingent signal, meaning that a loss only occurs if the signaller defaults from his honest signal. For example, in an aggressive encounter between two conspecifics, a weak animal may have a cost-free choice between signalling submission or signalling that it will fight. The latter option may be advantageous if the opponent is weak and submits as a result of the signal. This advantage may, however, be outweighed if it turns out that the opponent is strong and the signal results in a fight (cf. Enquist, 1985).
We have considered a simple example of an action-response game, but other more complex signalling games are possible. For examples of these and a general overview of signalling issues, see Silk et al. (2000), Hurd and Enquist (2005), and Számadó (2011).

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