Measured Games · Paper 07 of 07 · Behavioural ecology

Producer and scrounger

After C. J. Barnard and R. M. Sibly (1981); formalised by W. L. Vickery and colleagues (1991)

Abstract

Foragers choose between searching for resources and exploiting the discoveries of searchers. Because the return to exploiting falls as exploiters become common, the two payoffs cross once and the group settles at a stable interior proportion determined by the finder's share and group size alone. Free-riding is self-limiting wherever its payoff is negatively frequency dependent.

Keywordssocial foraging · negative frequency dependence · finder's share · stable mix

1Setup

Setup

Foragers in a group choose between two incompatible tactics. Producing means searching the environment for food patches. Scrounging means watching other group members and joining them when they find something. The tactics are incompatible because searching and watching demand different postures and different attention, which is what makes this a strategic choice rather than a level of effort.

The critical parameter is the finder’s share: the portion of a patch the discoverer consumes before anyone else arrives. A high finder’s share makes producing attractive. A low one makes it barely worth searching at all.

2The model

The model

# group size G, scrounger proportion q, patch value F # finder's share a, encounter probability λ per producer per round producer = λ [ aF + (1-a)F / (1 + qG) ] scrounger = λ (1-q)G · (1-a)F / (1 + qG) # setting them equal and solving q* = 1 - a - 1/G # stable: above q*, producing pays more and pulls q down # below q*, scrounging pays more and pushes q up

Neither λ nor F appears in the equilibrium. Only the finder’s share and the group size do, which is a sharp and testable prediction: change the layout of a feeding grid so that discoverers get a bigger head start, and the proportion of scroungers should move to a specific new value.

Figure 1Agent-based foraging with tactic switching

This figure needs JavaScript. The panels below are live simulations rather than images, so with scripting turned off they stay empty and the controls do nothing. Everything the figure demonstrates is also stated in the surrounding text: the model is given in full in section 2 and the result in section 3.

Flock · producers searching, scroungers watching

Scrounger proportion against the predicted equilibrium

Payoff curves · the crossing is the equilibrium. Hover to read intake at any mix

Producers Scroungers Food patch Predicted q* Producer intake Scrounger intake

Charts are focusable. With a chart focused, use the left and right arrow keys to read values along the horizontal axis.

Finder's share a

Group size G

Encounter rate λ

Imitation rate

Predicted q*—

Observed q—

Producer intake—

Scrounger intake—

Patches found—

Round0

Figure 1. Individuals switch tactic by copying whichever neighbour did better last round, so the proportion is an emergent count rather than a target. It converges on q* = 1 - a - 1/G from either direction: press Start all scroungers and it climbs down, press Reset and it climbs up. Agents explore at a rate of 0.5 per cent per round, which is what stops an all-scrounger flock from starving in place. Push the finder's share above roughly 1 - 1/G and scrounging stops paying at all, and the flock goes to pure producing.The lower panel is the diagram the model is usually stated with: producer and scrounger intake plotted against the proportion scrounging, crossing exactly once. Hover it to read both payoffs at any mix, and watch the observed marker slide onto the crossing as the flock settles.

3Results

Results

The two payoff curves cross once, and the crossing is a stable equilibrium maintained by negative frequency dependence. Any deviation is self-correcting. This is not a population split into two fixed types: it is a mixed strategy played by individuals who switch, and the observed proportion is the aggregate of that switching.

Vickery, Giraldeau, Templeton, Kramer and Chapman (1991) worked out the full model with a third option, opportunism, and established when each tactic can persist. Their central finding is that scrounging cannot take over the group, because its own success erodes as it spreads. The equilibrium is set by the finder’s share and group size, not by any capacity for enforcement, punishment or recognition.

That makes this the reference case worth keeping close to hand. Exploiting the effort of others does not by itself produce collapse. Whether it does depends entirely on whether the exploitative tactic’s payoff falls as it becomes common.

4Evidence

Evidence

Barnard and Sibly (1981) introduced the model alongside experiments on captive flocks of house sparrows (Passer domesticus). They manipulated flock composition and found that individuals adjusted their tactics in the direction the model predicts, with the frequency of scrounging responding to how many producers were present.

The decisive test came from Mottley and Giraldeau (2000), working with flocks of nutmeg mannikins (Lonchura punctulata). Rather than measuring a standing equilibrium and checking it for consistency, they manipulated the finder’s share directly by changing the design of the feeding grid, which moves the predicted equilibrium to a known new value. The flocks converged on the newly predicted proportion. That is a prediction followed by a test rather than a fit, and it is the reason this model is treated as well supported rather than merely plausible.

Giraldeau and Caraco (2000) collect the wider body of social foraging work, including applications to fish shoals and primate groups.

An independent case comes from outside foraging altogether. Brockmann, Grafen and Dawkins (1979) studied nesting in the digger wasp Sphex ichneumoneus, where females either dig their own burrow or enter one another wasp has abandoned. Entering is not a different kind of wasp: individuals do both, and the decision is not predicted by size, by date in the season or by an individual’s success. They tested mixed-strategy models against comprehensive field records from two North American populations, and the quantitative predictions were met in one population but not in the other. It is cited as one of the very few documented cases of a mixed evolutionarily stable strategy in nature, and the split result is as informative as the fit: the same model, the same species, two sites, one agreement and one failure.

5Limitations

Limitations

The model assumes scroungers detect every discovery and arrive in time to share it, that producers never scrounge, and that all group members are interchangeable. Real flocks violate all three. Detection is imperfect and distance-dependent, individuals mix tactics within a bout, and dominance rank strongly affects who gets a share of a patch, which means the aggregate equilibrium can be right while no individual is playing it.

There is also a measurement problem. The finder’s share is difficult to observe directly in the field and is usually inferred from patch geometry, so the sharpest tests have all been done in aviaries with engineered feeding grids. That is a real limitation on how far the nutmeg mannikin result generalises.

What survives all of that is the qualitative claim, which is well supported: the proportion is stable, it is set by how much of a find the finder keeps, and it does not require anyone to police anyone.

References

Brockmann, H. J., Grafen, A., & Dawkins, R. (1979). Evolutionarily stable nesting strategy in a digger wasp. Journal of Theoretical Biology, 77(4), 473–496. https://doi.org/10.1016/0022-5193(79)90021-3
Barnard, C. J., & Sibly, R. M. (1981). Producers and scroungers: A general model and its application to captive flocks of house sparrows. Animal Behaviour, 29(2), 543–550. https://doi.org/10.1016/S0003-3472(81)80117-0
Giraldeau, L.-A., & Caraco, T. (2000). Social foraging theory. Princeton University Press.
Giraldeau, L.-A., & Dubois, F. (2008). Social foraging and the study of exploitative behavior. Advances in the Study of Behavior, 38, 59–104. https://doi.org/10.1016/S0065-3454(08)00002-8
Mottley, K., & Giraldeau, L.-A. (2000). Experimental evidence that group foragers can converge on predicted producer–scrounger equilibria. Animal Behaviour, 60(3), 341–350. https://doi.org/10.1006/anbe.2000.1474
Vickery, W. L., Giraldeau, L.-A., Templeton, J. J., Kramer, D. L., & Chapman, C. A. (1991). Producers, scroungers, and group foraging. The American Naturalist, 137(6), 847–863. https://doi.org/10.1086/285197

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