Explication of the Cultural Transmission Model

by Elisa Jayne Bienenstock, Michael McBride
Explication of the Cultural Transmission Model
Elisa Jayne Bienenstock, Michael McBride
American Sociological Review
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Comment on Mark,ASR, June 2002

Explication of the ~dturalTransmission Model

Elisa Jayne Bienenstock Michael McBride

University of California, Iwine University of California, Iwine

n "Cultural Transmission, Disproportionate Prior Exposure, and the Evolution of Cooperation," Mark (2002) claims to have "con- ducted a formal theoretical analysis that isolat- ed the logic of cultural transmission [to] reveal an evolutionary force toward cooperation under conditions previously thought to make the evo- lution of cooperation impossible" (p. 324). The most striking aspect of Mark's work is not that his simple model converges to cooperative behavior, but that it works under initial condi- tions severely biased against the evolution of cooperation: randomly matched interaction in a large population with only 1% initial coopera- tors (99% initial defectors). Although this result is compelling at first glance, we show that it is not robust to minor changes in the model: coop- erative behavior evolves only when the partic- ular model specifications chosen by Mark are selected and does not evolve when the model is generalized. In particular, we show that the result relies on a decoupling between actors' fit- ness and behaviors. According to Mark's theo- retical model, this decoupling between fitness and behaviors should be associated, depends on simplifications of his own proposed mech- anism, and is not robust in small populations. We conclude that the role of Mark's formulation of disproportionate prior exposure (DPE) for the evolution of cooperation is unclear, though it

Direct correspondence to Elisa Jayne Bienenstock, Booz Allen Hamilton. Fall Church, VA 22042 (ejb@bah.com). We would like to thank Michael Macy, Alison Bianchi and two anonymous reviewers for valuable comments on earlier drafts.

may be a relevant factor in understanding the transmission of cultural traits like language.


Mark's thesis is that "disproportionate prior exposure creates a cultural evolutionary force toward cooperation" (p. 324). Language is the best illustration of how DPE is required for cul- tural transmission because all speakers of a lan- guage have been repeatedly exposed to the language. Mark's purpose is to extend DPE to cooperative behavior; he claims that "cultural transmission of almost any characteristic pro- duces disproportionate prior exposure to that characteristic7' (p. 330). This reasoning leads him to conclude that DPE must be present if a trait is present (footnote 12, p. 330).

To investigate this relation, Mark presents a computational model that randomly pairs mem- bers of a large population in one-shot prisoner's dilemma (PD) game. Figure 1 shows the pay- offs. Each individual has two traits: a strategy (i.e., propensity to either cooperate or defect) and a fitness level with values 0, 1, 2 or 3. Departing from most work in evolutionary game theory (Leimar and Hammerstein 200 1 ;Nowak and Sigmund 1998a; Nowak and Sigmund 1998b; Wilson 1989), Mark defines fitness as the value acquired during the previous pairing

Figure 1. Mark's Prisoner's Dilemma Game.

rather than the cumulative payoff to that indi- vidual. Cultural transmission only occurs when the paired actors have different fitness levels: after a pair engages and payoffs are awarded, the partner who entered the encounter with a lower fitness level will copy the behavior of the part- ner who entered the encounter with the higher fitness level. The change in strategy occurs after both actors have played their strategies (i.e., a low fitness partner cannot know and, therefore, cannot copy the strategy of a high fitness part- ner until strategies are revealed). After playing their strategies, actors retain the fitness they acquired into the next round.

Figure 2 illustrates how pairings lead to cul- tural adoption. The rows and columns list all possible combinations of strategies and fitness levels available to the actors. Cell values rep- resent the strategy and fitness level of the row player after an interaction with the column play- er. For example, if a Dl (row player) is paired with a C2 (column player), the Dl will becomes a C3: i.e., Dl earned a fitness level of 3 by defecting on a cooperator, while then adopting the C2's behavior because 2 > 1. This combi- nation of payoff structure and behavioral adop- tion dynamics converges to cooperative populations even when the initial population contains 99% defectors and 1% cooperators. Mark claims, "This model illustrates how the disproportionate prior exposure inherent to cul- tural transmission logically implies evolution- ary force toward cooperation "(p. 332).

Figure 2. Player strategy and fitness updating Matrix in Mark's Model.

* The presence of an attribution error that biases the
adoption of cooperation upward.
** The presence of an attribution error that biases the

adoption of defection downward. Adoption is enclosed in dark borders.


Mark further claims that his analysis "reveals the substantive reason cultural transmission favors the evolution of cooperation7' (pp. 3367). Defectors become cooperators only after inter- acting and benefiting from a cooperator's behav- ior. When a defector adopts cooperation, he maintains his high fitness level. As seen in Figure 2, this occurs when a DO or D 1 meets a C2 or C3 and becomes a C3. Conversely, when a CO meets a Dl or D3, or when a CO or C2 meets a D3, it becomes a DO and the average fit- ness level of defectors decreases. Over time, this process causes the relative proportion of coop- erators to increase because more actors benefit from meeting cooperators, who are more like- ly to have high fitness levels. Mark's explana- tion is correct on the surface, but it misses the more fbndamental and problematic question: Why does this model produce cooperation? A closer look reveals that more work is required to explain "the prevalence of cooperation in human populations7' (p. 325).


Mark's model converges to cooperation prima- rily because his mechanism for strategy switch- ing has a systemic asymmetric bias that is at odds with the theoretical motivation of his own proposed mechanism. To justify using the high fitness level as motivation for cultural adop- tion, Mark cites empirical work that shows that the behavior of high status individuals is more likely to be imitated than that of low status indi- viduals [Eisenberg-Berg and Geisheker 1979; Boyd and Richerson (1985); Katz and Lazarsfeld (1 955)l. Research not cited in Mark's study suggests that those with higher status are thought to be more competent (Webster and Foschi 1988), and when individuals of high sta- tus are shown to be less competent, then the level of their influence decreases (Wagner et al. 1996). Mark's implied assumption is that the observed fitness level and behavior are associ- ated, so the behavior of higher status actors is imitated. However, in Mark's model, fitness level and strategy are decoupled. When a DO or Dl meets a C3 with a high fitness level (that could have resulted only from a prior defec- tion), the DO or Dl copies the current behavior (cooperation) of this higher fitness partner, even though this fitness is the outcome of a prior defection. It is as if the DO or Dl incorrectly


attributes the high fitness to the observed behav- ior. Asterisks in the bottom left quadrant of Figure 2 mark instances of this type of "attri- bution error." If switching did not occur in these instances, there would be fewer conversions to cooperation. Double asterisks in the top left quadrant of Figure 2 mark a different but relat- ed situation: cooperators (CO and C2) remain cooperators after meeting more fit cooperators (C3) who achieved this high fitness level by defecting, thus slowing the defection rate.

Our concern is not that strategy and observed fitness are decoupledpev se, but that it is this decoupling that actually drives the convergence to cooperation. The success of Mark's model necessitates these biases, which only occur when an individual is matched with a fitter coopera- tor who benefited by defecting on another's cooperation. Limiting the certainty of decou- pling prevents the system from achieving coop- eration. Table 1 lists the minimum probability of a switch, when a low C or D and a C3 are paired, required to generate a cooperative equi- librium, at different starting population distri- butions. For example, for an initial population of 50% cooperators and 50% defectors to con- verge to cooperation, a decoupling of "status" and strategy must be present 49% of the time. When the initial proportion of population has 10% cooperators, the decoupling is required 98% of the time. The justification for using "fitness" as a trigger for adopting a new strat- egy is based on literature that links adoption of a behavior to status through an attribution that status is related to the behavior, but for con- vergence to cooperation, Mark's model requires that fitness and strategy not be connected. We find this contradiction problematic.


As stated earlier, one compelling feature of Mark's model is that, unlike many models, it converges to cooperation for large, randomly matched populations. Ironically, convergence to cooperation is not guaranteed in smaller pop- ulations unless the initial population has a suf- ficient proportion of cooperators. Mark models interaction using difference equations, which are good approximations for the dynamics of ran- dom interaction of strangers without exit options in very large (infinite) populations, but not so for modeling random matching in smaller (finite) populations. Suppose the initial popu- lation consists of n individuals of which k are COs and (n-k) are DOs. If the proportion of Cs is small and, by the randomness of the match- ing, all COs happen to be paired with DOs in the first round, then each CO will remain a CO, and each DO will either become a Dl or a D3. In the second round, unless the CO is paired with another CO, each CO will become a DO, each D matched with a CO will be a D3, while all other D3s become Dls. After only two rounds, the system eliminates all cooperators. This elimi- nation is more likely the smaller the population. The exact probability of the immediate elimi- nation is:

Table 2.la-c presents results from simula- tions with direct random matching consistent with the probabilities calculated above for n = 500, 5000, or 10,000, with k fixed at 1% of n (i.e., k = 0.01 *n) as in Mark's own benchmark. This shows that the probability of elimination grows as population size decreases, though an increase in the proportion of initial coopera- tors offsets the population size effect (as seen in Table 2.2a-c). So, although Mark's model does converge to cooperation when populations are large even when they are dominated by defectors, this convergence is unlikely to occur in small populations unless there is a suffi- ciently large proportion of cooperators.

Other evolutionary game theory models have only been able to achieve cooperation when

Table 1. Minimum Attribution Error Required to Converge to All-C Given an Initial Starting Population

Initial Population% %CO 1% 5% 10% 25% 50% 75% 99% %DO 99% 95% 90% 75% 50% 25% 1%

Minimum Attribution Error 0.9999 0.9954 0.9818 0.8776 0.4879 0.0480

Note: Calculated by modifying Mark's difference equations to account for attribution errors. Contact the authors for details.

Table 2. Convergence Results from Simulations with Actual Randomness

Mark's Model by Population Size One Permutation from Mark (2002)

(c) (4 (el (f, n = 10.000 Twice-in-row Cumulative

2 Period All Prior

2.1 Defector start (1% CO, 99% D3) Converged to All-C 8% 61% 85% 41% Converged to All-D 92% 39% 15% 59% Average rounds to convergence (SD) 10.1 (27.7) 85.3 (68.0) 121.6 (53.3) 63.6 (72.6) Min. IterationIMax. Iteration 31177 31264 31252 31187

2.2 Random start Converged to All-C 100% 100% 100% Converged to All-D 100% 100%


Average rounds to convergence (SD) 27.9 (5.2) 23.1 (2.8) 27.6 (1.2) 23.1 (2.8) 40.5 (5.7)
Min. IterationIMax. Iteration 14/49 17138 23/32 17138 31181

2.3 Cooperator start (99% CO, 1% D3) Converged to All-C Converged to All-D 100% 100% Average rounds to convergence (SD) 23.1 (2.8) 170.1 (14.5) Min. IterationiMax. Iteration 1713 8 135l245

Note: Convergence results obtained from simulation data generated by authors using 1000 runs per game configuration. The Java simulation code is available from the authors upon 5 request. All-C denotes "all donators" under altruism payoffs in column (d).






populations are small. Therefore, researchers have argued that, because humans evolved in rel- atively small groups (Leimar and Hammerstein 2001), models that allow for the evolution of cooperation in small groups are useful and may mimic how cooperation in human populations evolved. Finding a mechanism that fosters coop- eration in large populations is an achievement. However, its relevance for explaining the evo- lution of cooperation is ambiguous if it cannot also explain the evolution of cooperation in small populations, where cooperation is thought to have evolved.


The concerns above focus on Mark's model and should not be thought of as an evaluation of DPE, in general. In fact, Mark's model could better be described as a "higher status exposure" model, because only one exposure to a higher status individual is sufficient for strategy switch- ing. In this sense, Mark's model does not cap- ture DPE (p. 332), because in his model, the decision horizon is based only one interaction.

One way to capture the longer decision hori- zon implied by "disproportionate" prior expo- sure is by keeping the PD payoffs but requiring that an actor be matched with a fitter partner of a different strategy twice in a row, rather than once, to induce a conversion. As seen in Table 2(d), convergence to cooperation is now unat- tainable. A defector has to meet a fitter coop- erator twice in a row to switch, but after meeting a cooperator the first time, the defector becomes a D3. There is no possibility to meet a cooper- ator with fitness level higher than 3, therefore, this defector cannot be converted in the next round.

A second way to broaden the horizon of actors is to define the fitness level as the cumu- lative score obtained through more than one payoff. By encompassing an actor's history rather than the snap shot of one event, cumula- tive scores better represent the impact of strate- gies. Earlier work on the evolution of cooperation (cited above) focuses on cumulative scores for this reason. As shown in Figure 3a, if we begin with a random distribution of strate- gies all with a fitness level of zero, matches pro- duce four strategy payoff combinations. When these strategies are paired and their payoffs add to (Figure 3b), rather than replace (Figure 2), their prior fitness, there is no advantage to coop- erators (cooperators comprise fewer than half of the population). Figure 3c illustrates the fitness behavior pairings that are possible after the third iteration. Although the actual proportion of defectors depends on the random pairing process, this figure demonstrates the potential for defectors to gain high fitness levels relative to cooperators. Although not obvious from Figure 3, Table 2e-f illustrates that this process converges to defection. For Mark's benchmark of initially 99% defectors and 1% cooperators, extending the payoff window to two periods, results in a system that converges to coopera- tion only 41 % of the time. Further extending the payoff window to include all payoffs dooms cooperation, even with initially 99% coopera- tors and 1% defectors. Defectors can achieve high fitness levels rapidly, maintain their high fitness levels, and never meet fitter cooperators. More generally, a player's fitness could consist of the sum of payoffs from the last x rounds. Under this formulation, the chances of con- verging to cooperation diminish as x increases. A slightly different formulation would have a player's fitness be a discounted sum of past payoffs, where payoffs from the distant past are discounted more heavily. Again, increasing the discount factor decreases the chances of con- verging to cooperation.

While our analysis implies that Mark's model cannot account for the evolution of cooperation except when there is a large initial population

Figure 3. Player-strategy and fitness updating matrix when fitness is cumulative for three itera- tions.

of cooperators, the status switching mechanism may still be relevant in cultural transmission, where there is no decoupling of fitness and strategy. From a game theory perspective, lan- guage transmission is a coordination game and not a mixed motive (PD) game: both parties benefit only when speaking the same language, and adopting a new language is advantageous only if a partner speaks it. Additionally, lan- guage adoption is a learning process that requires repeated exposure so the extended time horizons implied by DPE are reasonable. However, in a PD game, copying the behavior of those from whom an actor has benefited is less likely to generate positive outcomes unless the population is sufficiently cooperative. Hence Mark's proposed mechanism may be useful for modeling the transmission of some cultural traits, but is not be as universal as Mark suggests.


Despite this model's sensitivity to randomness in small populations and dependence on short time horizons, Mark's work raises a number of interesting issues. The DPEIstatus model pro- vides valuable insights into a large class of interesting phenomena: cultural transmission arising in coordination games where the processes do not exhibit asymmetric attribu- tion biases. A systematic look at these biases may provide interesting insights into the dynamics of cultural transmission. Similar models, which focus more on disproportion than status as the criterion for transmission, might provide a contrasting result that sheds more light on the role of culture in the evolution of cooperation. Even if DPE is not an appropriate mechanism to model the evolution of cooperation, it is like- ly that culture will be an important component in the story of cooperation. For example, explic- itly using networks to study the effects of rep- utation on cooperation is another promising approach for studying cultural transmission.


Elisa Jayne Bienenstock is an Associate at the con- sultingJirm ofBoozAllen Hamilton where she applies mathematical sociology to national security issues. Currently her work focuses on designing methods to eliminate bias in task groups and developing pre- dictive models based on a combination of network analysis, decision analysis, and game theory.

Michael McBride is Assistant Professor of Economics at the University of California, Irvine. His current research uses game the0 y to study the for- mation of social networks and the causes ofpoliti- cal instability in sub-Saharan Africa. He is currently conducting laboratory experiments to test theoreti- calpredictions ofthe provision ofpublic goods when individuals do not know how many contributions are needed to provide the good. His article on statistically testing the relationship between income and subjec- tive well-being appeared in the Journal of Economic Behavior and Organization (2001).


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Nowak, M. A. and K. Sigmund. 1998a. "The dynam- ics of indirect reciprocity." Journal of Theoretical Biology. V194:561-574.

. 199813. "Evolution of indirect reciprocity by image scoring." Nature V393:573-578.

Wagner, David G., Rebecca S. Ford, andThomas W. Ford. 1996. "Can Gender Inequalities Be Reduced?" American Sociological Review 51:47-61.

Webster, Murray Jr. and Martha Foschi. 1988. "Overview of Status Generalization." Pp. 1-20 in

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edited by M. Webster, Jr. and M. Foschi. Stanford, California: Stanford University Press.

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