This inclination towards coherent beliefs contrasts sharply with foundationalist approaches to knowledge, which advocate building beliefs from self-evident truths (e.g., Descartes’ “I think; therefore, I am”). Our belief systems instead resemble intricate webs, characterised by complex interdependencies rather than clear hierarchies. Coherence-based reasoning is a cognitive model that attempts to capture these realities of human thought, where our beliefs are formed and held within this interconnected framework. This model not only describes how we form new beliefs but also how we maintain and adjust our existing belief systems in the face of new information.
Experimental evidence supports the key hypotheses of coherence-based reasoning, showing that incipient conclusions can change how evidence is weighted. For example, in an experiment where participants were asked to judge a legal case, their evaluation of the evidence shifted to align with their eventual conclusions (e.g., Fig. 1). This ‘coherence shift’ phenomenon underscores the bidirectional nature of human reasoning processes.
Coherence-based reasoning, a Gestaltian cognitive-consistency model of human reasoning (reviewed in Simon and Read (2023)), is founded on the observation that we tend to accept propositions that align with our existing beliefs and reject those that contradict them. This theory, as formalised by Thagard (1989) in the ECHO (Explanatory Coherence by Harmony Optimisation) model, offers a compelling alternative to unidirectional theories of reasoning such as Bayesian inference. Proponents of coherence-based reasoning, like Simon et al. (2004), argue that the theory’s ability to capture ‘coherence shifts’ – such as those observed in the legal judgement experiment described earlier — makes it a more accurate representation of human cognitive processes (see also: Simon (2004); Glöckner and Betsch (2008)).
The ECHO model, proposed by Thagard (1989), formalises coherence-based reasoning using a connectionist framework. This model can be simulated using neural network techniques (e.g., Fig. 2) and consists of the following elements:
This network structure allows the model to simulate the complex interplay between different pieces of information and beliefs, mirroring the web-like nature of human cognition. By observing how activation spreads through a network, we can gain insights into how people might integrate new information, resolve contradictions, and ultimately form coherent belief systems.
Consider the network structure illustrated in Fig. 3a, adapted from Fig. 8 of Thagard (1989). This network represents a scenario with five pieces of evidence (E1 to E5) and four hypotheses (H1 to H4). The relationships between these elements are as follows:
To simulate this model, activation propagates from the data to the evidence nodes, which in turn activate their connected hypothesis nodes. The nodes then influence each other based on their positive (solid lines) or negative (dashed lines) relationships. This process continues until the system reaches a steady state, representing the final decision regarding the believability of the hypotheses and evidences.
The simulation requires setting initial weights for the links between nodes. Following Thagard (1989), I used the default weightings:
These weightings may by modified to penalise the complexity of an explanation and to adjust the weighting of analogical links (no analogical links in this example).
The model is simulated as follows.
For each evidence and hypothesis node \(j\), calculate the net input to that node from all other nodes \(i\) connected to it
\[\text{net}_j = \sum_i w_{ij} a_i \label{E:net_j}\]For each evidence and hypothesis node \(j\), update the node’s activation level \(a_j\) according to
\[a_j(t+1) = a_j(t) (1 - \theta) + \begin{cases} \text{net}_j (\text{max} - a_j(t)) & \text{if } \text{net}_j > 0, \\ \text{net}_j (a_j(t) - \text{min}) & \text{otherwise.} \end{cases} \label{E:a_j_update}\]where max is the maximum node activation (default \(\text{max} = 1\)), min is the minimum node activation (default \(\text{min} = -1\)), and \(\theta\) is an activation decay parameter (default \(\theta = 0.05\)).
Fig. 3b illustrates how the activation levels of selected nodes in our example change over time. For example, the activation level of hypothesis H1 increases over time while all others H2-H4 decrease below zero. This represents the process of arriving at the conclusion that H1 is true and the other hypotheses false.
Notably, the example also illustrates the potential for evidence suppression. Although E5 is an established piece of evidence, its activation decreases over time. This occurs because E5 only supports the less favoured hypotheses H2-H4, which are ultimately rejected by the model.
Thagard (1989) explains the rationale between this counter-intuitive behaviour:
Principle 4 [of his 7 principles of explanatory coherence] asserts that data get priority by virtue of their independent coherence. But it should nevertheless be possible for a data unit to be deactivated. We see this both in the everyday practice of experimenters, in which it is often necessary to discard some the data because they are deemed unreliable (Hedges 1987), and in the history of science where evidence for a discarded theory sometimes falls into neglect (Lauden 1976).
While we often view evidence suppression as a cognitive flaw, the model and example above suggests it may be an emergent property of an otherwise adaptive mechanism. Our ability to discount or deactivate certain pieces of information, like E5 in the example above, enables us to form coherent, actionable beliefs in a world of noisy, often contradictory data. This process isn’t about wilfully ignoring evidence, but rather about contextualising it within a broader explanatory framework.
While this coherence-seeking tendency can be advantageous in many real-world scenarios, it may also underlie problematic cognitive biases such as confirmation bias or belief perseverance. Understanding this dual nature of coherence-based reasoning — its adaptive value and potential pitfalls — offers a rather nuanced view of human cognition, and may lead to more sophisticated models of reasoning and decision-making.
Glöckner, A. and Betsch, T. (2008). Modeling option and strategy choices with connectionist networks: Towards an integrative model of automatic and deliberate decision making, Judgment and Decision making 3(3): 215–228.
Simon, D. (2004). A third view of the black box: Cognitive coherence in legal decision making, University of Chicago Law Review 71: 511.
Simon, D. and Read, S. J. (2023). Toward a general framework of biased reasoning: Coherence-based reasoning, Perspectives on Psychological Science p. 17456916231204579.
Simon, D., Snow, C. J. and Read, S. J. (2004). The redux of cognitive consistency theories: evidence judgments by constraint satisfaction, Journal of Personality and Social Psychology 86(6): 814.
Thagard, P. (1989). Explanatory coherence, Behavioral and Brain Sciences 12(3): 435–467.
]]>Many fishes perform predator-inspection, which is when a small number of individuals will break away from the main group and approach a potential predator (Dugatkin, 1988). Performing the inspection is risky behaviour, but it also benefits the group; it can provide the information needed to either escape or return foraging if there’s no real threat (Padget et al., 2023). The ideal situation for any individual is for the inspection to take place but to not be one of the volunteers (but see Godin and Davis, 1995).
This kind of situation is modelled by theorists using the Volunteers’ Dilemma (e.g., Archetti, 2009), which is a public goods game where the good is only produced if at least one group member volunteers to pay the cost. In the case of guppies, the ‘public good’ may be the information, and the cost is the risk of being eaten. This kind of game can be contrasted with the more familiar case of the (linear) public goods game, where the amount of public good increases linearly with the amount of cooperation. I’m particularly interested in this kind of model because, in contrast to the linear public good, nonlinear games like the Volunteers’ Dilemma have the possibility evolving and/or sustaining cooperation without assortment (Archetti et al., 2020) (though guppies might assort; see Darden et al. (2020)). Given that almost nothing is linear in biology, I think nonlinearity might have implications for how cooperation evolved (Kristensen et al., 2022).
Volunteers’ Dilemma models predict that volunteering will decrease as the group size increase– but Padget et al. (2023) found the opposite. Comparing three group sizes, they observed that “Individuals in large groups inspected more frequently than those in both intermediate groups (moderate evidence) and small groups (weak evidence)” (Fig. 1). This suggests the models are missing something. They discussed a variety of ways the models could be made more realistic and whether or not such models could reproduce the observed behaviour (Supplement, my notes here), but the short of it was that the question remains open.
As an exercise to learn more about mixed-strategy models, I created my own small model to explore. I focused on two aspects. First, the cost of volunteering to do the inspection is shared among the volunteers, so it should decrease as the number of volunteers increases. This sounds like it could potentially produce the desired positive relationship between volunteering and group size, but as Padget et al. (2023) point out, we already know that’s not the case. Weesie and Franzen (1998) investigated one such model, and they found that even if costs are shared between volunteers, volunteering will have a negative relationship with group size. Second, the cost of ignoring the potential predator is also likely shared among group members. This won’t produce the desired relationship, either, but I thought it would be a good exercise to combine the two.
Consider two strategies: \(V\), volunteer; and \(I\), ignore the potential predator and don’t volunteer. Consider a focal individual \(0\) who pursues strategy \(s_0 \in \lbrace V, I \rbrace\) Let \(k\) denote the number of volunteers among the nonfocal members. Then the payoffs to the focal are
\[\begin{equation} w_0 = \begin{cases} 1 - \frac{c}{k+1} & \text{if } s_0 = V, \\ 1 - \frac{a}{N} & \text{if } s_0 = I \text{ and } k = 0, \\ 1 & \text{if } s_0 = I \text{ and } k \geq 1 \end{cases} \end{equation}\]Denote the probability of ignoring a potential predator by \(\gamma\). The expected payoff from ignoring is
\[\begin{equation*} W_I(\gamma) = \underbrace{\gamma^{N-1} (1 - (a/N))}_{k = 0} + \underbrace{1 - \gamma^{N-1}}_{k \geq 1} = 1 - \frac{a \gamma^{N-1}}{N}, \end{equation*}\]where \(N\) is the group size. The expected payoff from volunteering is
\[\begin{align*} W_V(\gamma) &= 1 - c \sum_{k=0}^{N-1} \left( \frac{1}{k+1} \right) \binom{n-1}{k} (1 - \gamma)^k \gamma^{n-1-k}, \\ &= 1 - \frac{c}{N(1-\gamma)} (1 - \gamma^N). \end{align*}\]Therefore, the payoff to an individual pursuing strategy \(\gamma'\) in a population pursuing strategy \(\gamma\) is
\[\begin{equation} W(\gamma'; \gamma) = \gamma' W_I(\gamma) + (1 - \gamma') W_V(\gamma) = W_V(\gamma) + \gamma' \underbrace{(W_I(\gamma) - W_V(\gamma))}_{g(\gamma)} \end{equation}\]Following, e.g., Bach et al. (2006), the mixed Nash equilibrium occurs when the payoff to the focal player is the same regardless of the strategy probability it pursues. This occurs when \(\gamma = \gamma^*\) satisfies
\[\begin{equation} g(\gamma) = W_I - W_V = \frac{c (1 - \gamma^N) - a (1 - \gamma) \gamma^{N-1}}{N (1 - \gamma)} = 0. \end{equation}\]We have
\[\begin{align*} g(\gamma = 0) &= \frac{c}{N}, \\ g(\gamma = 1) &= c - \frac{a}{N}, \end{align*}\]where the second equation makes use of \(1 - \gamma^N = (1 - \gamma) (\gamma^{N-1} + \gamma^{N-2} + \ldots + 1)\). Therefore, provided \(c < a/N\), \(g(\gamma)\) has at least one root between 0 and 1.
To characterise the root further, consider the numerator of \(g(\gamma)\)
\[\begin{equation} u(\gamma) = c (1 - \gamma^N) - a (1 - \gamma) \gamma^{N-1}. \end{equation}\]At the ends of the function, \(u(\gamma=0) = c\) and \(u(\gamma=1) = 0\). The derivative of the numerator
\[\begin{equation} u'(\gamma) = \gamma^{N-2} \left[ \gamma N (a - c) - a (N - 1) \right]. \end{equation}\]The derivative changes sign exactly once in the interval \(\gamma \in [0, 1)\), at
\[\begin{equation} \gamma_{\text{sign change}} = \frac{a(N-1)}{N(a-c)}. \end{equation}\]Furthermore, provided that \(c < a/N\) (which is the condition for the existence of at least one root, as above), then \(u'(\gamma = 1) > 0\). Therefore, \(g(\gamma)\) has a single root \(\gamma^*\), which is to the left of the sign change
\[\begin{equation} \gamma^* < \frac{a(N-1)}{N(a-c)}, \end{equation}\]and at that root
\[\begin{equation} g'(\gamma^*) < 0. \end{equation}\]The strategy \(\gamma^*\) is an evolutionarily stable strategy (sensu Maynard Smith and Price) if, for small \(\varepsilon\)
\[\begin{equation*} \Delta W = W(\gamma^*; \varepsilon \gamma + (1 - \varepsilon) \gamma^*) - W(\gamma; \varepsilon \gamma + (1 - \varepsilon) \gamma^*) > 0 \end{equation*}\]is satisfied for all \(\gamma \neq \gamma^*\). We have
\[\begin{equation*} W(\gamma^*; \varepsilon \gamma + (1 - \varepsilon) \gamma^*) = W_V(\varepsilon \gamma + (1 - \varepsilon) \gamma^*) + \gamma^* g(\varepsilon \gamma + (1 - \varepsilon) \gamma^*) \end{equation*}\]and
\[\begin{equation*} W(\gamma; \varepsilon \gamma + (1 - \varepsilon) \gamma^*) = W_V(\varepsilon \gamma + (1 - \varepsilon) \gamma^*) + \gamma g(\varepsilon \gamma + (1 - \varepsilon) \gamma^*) \end{equation*}\]Therefore,
\[\begin{equation} \Delta W = (\gamma^* - \gamma) g(\varepsilon \gamma + (1 - \varepsilon) \gamma^*) = (\gamma^* - \gamma) g(\gamma^* - \varepsilon(\gamma^* - \gamma)) \end{equation}\]We perform a Taylor expansion around \(\gamma^*\)
\[\begin{equation*} g(\varepsilon \gamma + (1 - \varepsilon) \gamma^*) = g(\gamma^*) - \varepsilon (\gamma^* - \gamma) g'(\gamma^*) + \mathcal{O}(\varepsilon^2) \end{equation*}\]So
\[\begin{equation} \Delta W = (\gamma^* - \gamma) g(\gamma^*) - \varepsilon (\gamma^* - \gamma)^2 g'(\gamma^*) + \mathcal{O}(\varepsilon^2) \end{equation}\]At the mixed Nash equilibrium, \(g(\gamma^*) = 0\); therefore, in order to satisfy \(\Delta W > 0\), we must satisfy \(g'(\gamma^*) < 0\), which we obtained above.
We obtain the relationship between the ignoring probability at the ESS and the parameters and group size by implicitly differentiating \(u(\gamma^*)\).
The ignoring probability increases with cost of volunteering
\[\begin{equation*} \frac{d \gamma^*}{dc} = \frac{ (\gamma^*)^N - 1 }{ (\gamma^*)^{N-2} [\gamma^* N (a - c) - a (N - 1)] } > 0, \end{equation*}\]decreases with cost of being attacked
\[\begin{equation*} \frac{d \gamma^*}{da} = \frac{ \gamma^* (1 - \gamma) }{ \gamma^* N (a - c) - a (N - 1) } < 0, \end{equation*}\]and increases with group size
\[\begin{equation*} \frac{d \gamma^*}{d N} = \frac{ \gamma^* \ln(\gamma^*) [a (1 - \gamma^*) + c \gamma^*] }{ [\gamma^* N (a - c) - a (N - 1)] } > 0. \end{equation*}\]The final result, \(d \gamma^* / dN > 0\), agrees with the point Padget et al. made: sharing the costs does not change the direction of relationship between volunteering and group size predicted by models, which predict that volunteering decreases with group size.
One thing that might be missing from these models is coordination. For example, when pairs of guppies perform the inspection, they do so in a tit-for-tat-like manner, each continuing their approach towards the potential predator provided that the other also continues their approach (Dugatkin, 1988; Dugatkin and Alfieri, 1991). This coordinative behaviour could be generalised to larger inspection parties.
Archetti, M. (2009). Cooperation as a volunteer’s dilemma and the strategy of conflict in public goods games, Journal of Evolutionary Biology 22(11): 2192–2200.
Bach, L. A., Helvik, T. and Christiansen, F. B. (2006). The evolution of n-player cooperation—threshold games and ESS bifurcations, Journal of Theoretical Biology 238(2): 426–434.
Darden S. K., James R., Cave J. M., Brask J. B., Croft D. P. (2020). Trinidadian guppies use a social heuristic that can support cooperation among non-kin. Proc Biol Sci. 287(1934):20200487.
Dugatkin, L. A. (1988). Do guppies play tit for tat during predator inspection visits?, Behavioral Ecology and Sociobiology 23: 395–399.
Dugatkin, L. A. and Alfieri, M. (1991). Tit-for-tat in guppies (poecilia reticulata): the relative nature of cooperation and defection during predator inspection, Evolutionary Ecology 5(3): 300–309.
Godin J. J. and Davis S. A. (1995). Who dares, benefits: predator approach behaviour in the guppy (Poecilia reticulata) deters predator pursuit, Proc. R. Soc. Lond. B.259193–200.
Kristensen, N. P., Ohtsuki, H. and Chisholm, R. A. (2022). Ancestral social environments plus nonlinear benefits can explain cooperation in human societies, Scientific Reports 12: 20252.
Padget, R. F., Fawcett, T. W. and Darden, S. K. (2023). Guppies in large groups cooperate more frequently in an experimental test of the group size paradox, Proceedings of the Royal Society B 290(2002): 20230790.
Weesie, J. and Franzen, A. (1998). Cost sharing in a volunteer’s dilemma, Journal of Conflict Resolution 42(5): 600–618.
]]>We compiled a comprehensive dataset of biodiversity in Singapore, comprising more than 50,000 individual records and representing more than 3,000 species and ten major taxonomic groups (mammals, birds, reptiles, amphibians, freshwater fishes, butterflies, bees, phasmids, decapod crustaceans, and plants). We developed novel statistical methods that account for “dark extinctions”, i.e., extinctions of undiscovered species, and estimated that 37% of Singapore’s species have gone extinct over the last two centuries. We extrapolated Singapore’s historical experience to a future scenario for the whole of Southeast Asia and estimated that around 18% of species would be lost regionally by 2100.
Our extinction estimates for both Singapore and future Southeast Asia are a factor of two lower than previous estimates, which we attribute to our improved statistical methods. However, extinctions in Singapore have been concentrated among charismatic species, including large mammals, forest-dependent birds, butterflies, and orchids. Thus, we speculate that if deforestation continues across the region, Southeast Asia may come to resemble a “tropical Europe”, where a large majority of species persist in a human-dominated landscape, but where many of the most charismatic species are absent. We recommend that future tropical conservation efforts focus on charismatic species such as tigers, elephants, rhinoceroses and orangutans, in line with the classic umbrella species approach.
The paper has just been published in PNAS. It represents the culmination of a decade of work by many people to assemble the comprehensive database of Singapore biodiversity records and develop the novel statistical methods needed to account for dark extinctions.
Chisholm, R. A., N. P. Kristensen, F. E. Rheindt, K. Y. Chong, J. S. Ascher, K. K. P. Lim, P. K. L. Ng, D. C. J. Yeo, R. Meier, H. H. Tan, X. Giam, Y. S. Yeoh, W. W. Seah, L. M. Berman, H. Z. Tan, K. R. Sadanandan, M. Theng, W. F. A. Jusoh, A. Jain, B. Huertas, D. J. X. Tan, A. C. R. Ng, A. Teo, Z. Yiwen, T. J. Y. Cho, and Y. C. K. Sin. 2023. Two centuries of biodiversity discovery and loss in Singapore. Proceedings of the National Academy of Sciences 120:e2309034120
]]>Unfortunately, the fishing game was unexpectedly unavailable this year, but I was able to find a very nice last-minute alternative.
Charles Holt at the University of Virginia has made available the Veconlab software, a collection of 50 web-based experiments ranging from public goods games to auctions.
I just wanted to create a basic linear public goods game, and I was easily able to set one up using the adjustable parameter settings. The game was responsive and performed well during the tutorial. Students were incentivised by the promise of a conversion from dollars earnt to participation marks, and by 10 rounds, we had obtained the classic result of declining contributions.
]]>I was one of four presenters on the theme Modelling Our Changing Biosphere. I taught about evolutionary game theory, and we also learnt about modelling fisheries (with Nokuthaba Sibanda from University of Wellington), reinforcement learning (with Carl Boettiger from University of California), model sloppiness (with Matthew Adams from Queensland University of Technology), and sequential Monte Carlo methods (with visitors from the concurrent SMC Downunder Workshop).
My goal for the 9 hours was to introduce classical game theory, Maynard Smith ESS, replicator dynamics, and group games. The latter gave me an opportunity to share new theory I recently learnt about the connection between evolutionary stability in nonlinear public goods games and Bernstein polynomials (see blog post here).
It was important to me that a student with no background in either game theory nor evolutionary dynamics would be able to follow along, so I created a hybrid-style series of lectures where I would talk for a while, and then we would play a game-theoretic game or the students would try to solve an exercise using the techniques I just presented.
I took inspiration from a series of lectures by Ben Polak created for the Open Yale Game Theory course. I stumbled across them a few years ago in podcast form, and I would listen to them in the mornings as I rode to work. His lectures are so clear that, even without the video to see what he wrote on the board, I was able to follow what was happening.
We had a really enthusiastic bunch of students. One student, Christopher Brown, even coded up an Axelrod-style tournament for us to play during the fortnight. He added the twist that the tournament would run in an evolutionary manner, with the abundance of each strategy changing depending upon its payoff in the previous generation.
We had some excellent student presentations on a diverse range of topics.
I had a lot of fun teaching, and I also learnt a lot from attending the other courses and presentations. I highly recommend volunteering for an AMSI Winter School; it is very rewarding.
Here’s some feedback I received at the end of the course.
Nadiah’s game theory course was fascinating, and she made the process of learning very interactive by constantly asking the audience questions and performing trial runs of games through us.
Nadiah and Matthew were standout speakers. Having an audience with different levels of exposure to a topic is difficult but they made specific effort to ensure that everyone could understand their material.
Nadiah’s evolutionary game theory were fascinating and very engaging. It’s a topic I’ve read about, and found very interesting, before, but she still kept it engaging throughout, and thought parts of it that I hadn’t encountered before.
Nadiah has such a pleasant way of teaching that was really engaging.
Nadiah was the best from my point of view, some of what I was looking for is related to some of the approaches I have been considering. This presenter allowed time for people to work through the problems and then explained the steps through the problems.
Dr Nadiah Kristensen made game theory really approachable and engaging.
A special note for Nadiah who made the game theory course interesting and engaging
]]>I appreciated how well structured and inclusive Dr Nokuthaba Sibanda and Dr Nadiah Kristensen’s courses were - the lectures and exercises were thoughtfully designed, easy to follow along and started from first principles.
We’ve been putting the final touches on our extension of the higher-order genetic associations approach, to extend the scenario to games with many different strategies, and I’ll be here for another month so we can explore some new ideas we have for modelling homophilic group formation.
Hisashi’s group is interested in all kinds of theoretical problems in ecology and evolution, so I gave a talk showcasing some of the conservation/biodiversity work we do in the Chisholm Lab.
Apart from that, I brought my Brompton with me, and I’ve been having a lot of fun on the weekends exploring the bay area.
]]>Consider, for example, the method used by Australian Aborigines in southwestern Victoria to hunt kangaroos. As discussed in Balme (2018), in the early nineteenth century, communal gatherings were associated with mass-scale hunting of kangaroos and emus. People would form a large circle 25-30 km in diameter. Then they would then move inwards, yelling to frighten the animals, until they were concentrated in a small area where they could be killed. Obviously, these were modern hunter gatherers, and the hunts were just one small part of the activities that attracted participants from hundreds of kilometres away. Nevertheless, we can imagine that our much earlier ancestors might have struck upon a similar but much smaller-scale method of hunting.
Let’s consider what the benefits function of this encircling technique might look like. A single individual cannot successfully encircle the animals. As the number of hunters increases, at first the likelihood of success is small, but the likelihood increases at an increasing rate. However, once the number of hunters reaches a certain level, the animals are already surrounded, and additional hunters become increasingly superfluous. Therefore, we can expect something like a sigmoid relationship between the number of hunters and the benefit. In the most extreme case, the benefits function would have a threshold shape: a minimum number of hunters is needed to surround the prey, and beyond that, additional hunters make no difference (Fig. 2). This threshold game is an \(n\)-player generalisation of the 2-player Stag Hunt.
We expect that a group who hunts together will include some family members; however, modelling kin selection in groups is difficult when the benefits function is nonlinear. In short, unlike 2-player interactions, where the model can parameterised with single relatedness factor (i.e., Hamilton’s \(r\)), in group interactions, the model must account for all possible combinations of types within the group, which means accounting for all possible kin + nonkin combinations. For a more detailed discussion of this, see Allen & Nowak (2016), particularly Eq. 6; and Van Cleve (2015), particularly Eq. 21.
Hisashi Ohtsuki gave a recipe for describing the dynamics in terms of those kin + nonkin combinations (Ohtsuki, 2014). Let \(a_k\) and \(b_k\) be the payoff functions for Cooperators and Defectors, respectively, when \(k\) of the other \(n-1\) group members are Cooperators. Then the change in the proportion of Cooperators \(p\) in the population is
\[\begin{equation} \Delta p \propto \sum_{k=0}^{n-1} \sum_{l=k}^{n-1} (-1)^{l-k} {l \choose k} {n-1 \choose l} \left[ (1-\rho_1) \rho_{l+1} a_k - \rho_1(\rho_l - \rho_{l+1}) b_k \right]. \tag{1} \end{equation}\]where
\[\begin{equation} \rho_l = \sum_{m=1}^l \theta_{l \rightarrow m} p^m. \label{rho_l} \tag{2} \end{equation}\]The \(\theta_{l \rightarrow m}\) in Eq. 2 are called the higher-order relatedness coefficients, and they can be thought of as generalisations of the dyadic relatedness coefficient, Hamilton’s \(r\), to \(l\) individuals. Relatedness \(r\) is the probability that, if we draw 2 individuals without replacement from the group, they will share a common ancestor and therefore their strategy will be identical by descent. \(\theta_{l \rightarrow m}\) is the probability that, if we draw \(l\) individuals without replacement from the group, they will share exactly \(m\) common ancestors.
However, Ohtsuki (2014) did not provide a general method for obtaining the \(\theta_{l \rightarrow m}\) parameter values. In our paper, we created 3 homophilic group-formation models that allowed us to parameterise Eqns. 1 and 2 (Fig. 3). In short, we assumed that groups are formed sequentially by current members attracting/recruiting new members. We modelled homophily (the tendency to attract similar others) as an exogenous parameter, and when homophily is high, new members are more likely to be kin of an existing member and thus identical by descent.
We were interested in the question of how cooperation in non-linear games first arose. We know that, for a benefits function shaped like the hunting examples above, provided the function and game have suitable parameter values, then once the number of cooperators in the population is high enough, a coexistence between cooperators and defectors is evolutionarily stable. Peña et al. (2014) provides a particularly beautiful way to analyse this mathematically. However, the ancestral state was presumably a population of all defectors, and it can also be shown cooperators cannot invade a population of all defectors, which raises the question of how cooperation got started in the first place.
To understand why cooperators can persist but cannot invade, we can think about the one-shot threshold game as an example (Fig. 2). If you are in a group with complete strangers, you have no kinship or friendship incentives to cooperate. If the interaction is anonymous, you have no reputational incentives. If the game is a one-shot interaction, you know that you will never meet these people again, so you have no reciprocity / fear-of-punishment incentives, either. Nevertheless, it can still make sense to cooperate in a threshold game. If others in your group are likely to be cooperators, then cooperation may be in your self interest because your contribution might be the one to push the public-goods benefit above the threshold. Thus, cooperation can persist evolutionarily. However, if cooperation is very rare in the population such that you’re likely to be the only cooperator in the group, then it doesn’t make sense to cooperate because your lone contribution will not be enough to obtain the public good. Therefore, the first cooperators could never gain a foothold in the population and thus cooperation could never evolve.
However, the reasoning above assumes groups are formed at random with non-family members / strangers. In our paper, we showed that if groups in the past tended to include family members, then cooperation could evolve. If instead of being grouped with strangers, you are in a group with family members, then if you are a cooperator, your fellow group members are likely to be cooperators as well because they share your genes. This positive assortment between cooperators means that cooperation can gain a foothold in the population, and cooperation can evolve.
In addition, once cooperation has evolved by kin selection, then even if circumstances change and recruitment of strangers becomes more common, cooperation can nevertheless persist. Whether or not it persists depends on how much homophily is lost and the parameter values in the game (Fig. 4), but it is possible for cooperation to persist in some circumstances even if homophily is lost altogether (Fig. 4(b)).
So why do people cooperate in lab-based public goods games? A lot of attention has been paid to the fact that lab-based games are unrealistic: we often interact with people we will meet again and we are rarely truly anonymous, so the social heuristics we use for daily life lead us astray in this artificial environment. We emphasise that another way lab games are unrealistic is that people are not used to playing linear games, and we speculate that this might explain some of the cooperative behaviour.
There is some evidence supporting the idea that people in lab-based games behave as though they’re playing a nonlinear game. People generally prefer to condition their contributions on the level of contribution from others (Fischbacher et al 2001, Chaudhuri 2011, Thöni & Volk 2018), which only makes sense from a self-interested perspective if the game is nonlinear. They will even do this when playing against a computer, which suggests that this behaviour isn’t purely about a sense of fairness or caring for the welfare of others (Burton-Chellew et al. 2016). Chat logs from computer-networked games also reveal a common misperception of linear games as some type of coordination problem (Cox & Stoddard 2018) like a threshold game. Thus, it seems likely that some people are just genuinely confused about what the self-interested payoff-maximising strategy is when the game is linear, and they expect payoffs similar to a nonlinear game.
Our idea that people cooperate because they are “confused” is similar to the evolutionary maladaptionhypothesis; however, there are some key differences, as well. Roughly speaking, the evolutionary maladaptation hypothesis is the idea that when humans cooperate with strangers, we basically do so because our behavioural programming mistakes those strangers for kin (Burnham et al. 2005, Hagen & Hammerstein 2006, El-Mouden et al. 2012). For most of our evolutionary history, we have lived in small groups composed mostly of relatives. In that environment, indiscriminately cooperating with others around you was a good strategy because chances were they were your relatives. However, our social environment has changed very rapidly recently, in evolutionary terms, so that now we often interact with nonkin, and evolution hasn’t yet had a chance to catch up. Thus our behaviour is “maladaptive” because cooperating with relatives provided inclusive fitness benefits, whereas cooperating indiscriminately with strangers does not.
In contrast, in our model, cooperating with strangers is not maladaptive. Although past kin selection is needed to explain how cooperation first got started, once it is established in the population, cooperating with strangers can be in one’s self interest.
Our model also implies a different narrative about how/why cooperation persisted as humans transitioned from mostly interacting with family to interacting with nonkin. In the evolutionary maladaptation hypothesis, cooperation was extended to nonkin maladaptively due to, e.g., rapidly increasing population size. This seems to imply somehow that cooperating with nonkin was an “easy” mistake to make. In our model, cooperation persisted because it was evolutionarily stable. We expect that it might have been quite challenging to extend cooperation to nonkin because that means overcoming kin bias and a suspicion of strangers or outgroup members. Our view seems to sit more easily with cross-cultural empirical studies showing that cooperative behaviour with strangers/nonkin is not as universal in small-community societies as it is in societies with high market integration, etc. (Henrich et al. 2005, Henrich et al. 2010).
Our work is published now in Scientific Reports, and we are currently working on extending these modelling techniques from situations with only 2 strategies to many strategies (Cooperate, Defect, Coordinate, Punish, etc.).
Allen, B. and Nowak, M. A. (2016). There is no inclusive fitness at the level of the individual, Current Opinion in Behavioral Sciences 12:122-128.
Balme, J. (2018). Communal hunting by Aboriginal Australians: Archaeological and ethnographic evidence. In Manipulating Prey: Development of Large-Scale Kill Events Around the Globe, eds Carlson, K. & Bemet, L., University of Colorado Press, Boulder, Colorado, 42062.
Burnham, T. C. & Johnson, D. D. (2005). The biological and evolutionary logic of human cooperation. Anal. Kritik 27, 113-135.
Burton-Chellew, M. N., El Mouden, C. & West, S. A. (2016). Conditional cooperation and confusion in public-goods experiments. Proceedings of the National Academy of Sciences 113:1291-1296.
Chaudhuri, A. (2011). Sustaining cooperation in laboratory public goods experiments: A selective survey of the literature. Exp. Econ. 14, 47-83.
Cox, C. A. & Stoddard, B. Strategic thinking in public goods games with teams. J. Public Econ. 161, 31–43 (2018).
El-Mouden, C., Burton-Chellew, M., Gardner, A. & West, S. A. (2012). What do humans maximize? In Evolution and Rationality: Decisions, Cooperation and Strategic Behaviour, eds Okashi, S. & Binmore, K., Cambridge University Press, Cambridge, 23-49.
Fischbacher, U., Gächter, S. & Fehr, E. (2001). Are people conditionally cooperative? Evidence from a public goods experiment. Econ. Lett. 71, 397-404.
Hagen, E. H. & Hammerstein, P. (2006). Game theory and human evolution: A critique of some recent interpretations of experimental games. Theor. Popul. Biol. 69, 339-348.
Henrich, J. et al. (2005) “Economic man” in cross-cultural perspective: Behavioral experiments in 15 small-scale societies. Behav. Brain Sci. 28:795-815.
Henrich, J., et al. (2010). Markets, religion, community size, and the evolution of fairness and punishment. Science, 327(5972):1480-1484.
Ohtsuki, H. (2014). Evolutionary dynamics of n-player games played by relatives. Philosophical Transactions of the Royal Society B: Biological Sciences 369, 20130359
Peña, J., Lehmann, L. & Nöldeke, G. (2014). Gains from switching and evolutionary stability in multi-player matrix games. Journal of Theoretical Biology 346:23-33.
Thöni, C. & Volk, S. (2018). Conditional cooperation: Review and refinement. Econ. Lett. 171, 37-40.
Van Cleve, J. (2015). Social evolution and genetic interactions in the short and long term, Theoretical Population Biology 103:2-26.
]]>For this task, we will consider the replicator dynamics of a 2-player game with 3 strategies.
Recall the general equation for the replicator dynamics is
\[\begin{equation} \dot{p}_i = p_i (f_i - \bar{f}) \end{equation}\]where \(p_i\) is proportion of \(i\)-strategists in the population, \(f_i\) is fitness effect of strategy \(i\), and \(\bar{f}\) is the average fitness in the population.
The fitness effect is the expected payoff
\[\begin{equation} f_i = \sum_j p_j \pi(i \mid j) \end{equation}\]where \(\pi(i \mid j)\) is the payoff to an \(i\)-strategist who has been paired against a \(j\)-strategist. The average fitness
\[\begin{equation} \bar{f} = \sum_j f_j p_j. \end{equation}\]For this example, the payoffs are given by the matrix
\[\begin{equation} \pi = \begin{pmatrix} 0 & 1 & 4 \\ 1 & 4 & 0 \\ -1 & 6 & 2 \\ \end{pmatrix}, \end{equation}\]where \(\pi(i \mid j)\) is given by element \(\pi_{i,j}\). For example, when strategy 1 plays against strategy 2, strategy 1 receives payoff 1; when strategy 1 plays against strategy 3, strategy 1 receives payoff 4; and so on. This example is taken from Ohtsuki et. al. (2006).
To gain an intuition for the dynamics, we will first plot them.
Marvin Böttcher has written a handy utility for plotting the dynamics of a 3-strategy game on a simplex called egtsimplex. It can be downloaded from the Github repository here: https://github.com/marvinboe/egtsimplex. The repository includes an example that we can look at to see how it works.
First, we need to create a function that defines the dynamics and returns \(\dot{\boldsymbol{p}}\).
Above, I chose to define each fitness effect in a for-loop to be as explict as possible about the connection to the equations above.
To plot the dynamics, we create a simplex_dynamics
object called dynamics
, and use the method plot_simplex()
:
That produces a nice graph like the one below.
In the process of finding the dynamics, egtsimplex stored the fixed points it found…
… however, these are in \((x,y)\) coordinates for plotting.
To get the barycentric coordinates, i.e., the actual quanties of \(p_1\), \(p_2\), and \(p_3\),
we can use the xy2ba()
method
The interior steady state is the 3rd one in fp_ba
above.
We can already tell from the plot there are oscillatory dynamics,
but it’s not immediately obvious whether the interior steady state is an attractor, repellor, or neutral.
Let’s try plotting a trajectory that starts nearby.
To plot a trajectory, I’ll use solve_ivp
with the LSODA
method.
I know that \(\sum_j p_j = 1\), so I’ll write a lambda function that takes into account that constraint
and reduce the number of state variables from 3 to 2.
To add the trajectory to the plot,
we will need to revert pt
above from a 2-dimensional to 3-dimensional system,
and then convert the 3-dimensional barycentric coordinates to \((x,y)\) for plotting.
That produces the figure below, which suggests that the interior steady state is unstable.
Ohtsuki et al. (2006) determined the eigenvalues of the Jacobian matrix analytically, but let’s use this example to learn how we would do so numerically.
First, we need to get an expression for each element of the Jacobian matrix.
Recall the replicator dynamics:
\[\begin{equation} \dot{p}_i = p_i (f_i - \bar{f}) \end{equation}\]Each element of the Jacobian matrix
\[\begin{equation} J_{i,k} = \left. \frac{\partial \dot{p}_i}{\partial p_k} \right|_{\boldsymbol{p}^*} = \left. [f_i - \overline{f}] \right|_{\boldsymbol{p}^*} + p_i^* \left( \left. \frac{\partial f_i}{\partial p_k} \right|_{\boldsymbol{p}^*} - \left. \frac{\partial \overline{f}}{\partial p_k} \right|_{\boldsymbol{p}^*} \right) \end{equation}\]At \(\boldsymbol{p}^*\), \(f_i = \overline{f}\), so
\[\begin{equation} J_{i,k} = p_i^* \left( \left. \frac{\partial f_i}{\partial p_k} \right|_{\boldsymbol{p}^*} - \left. \frac{\partial \overline{f}}{\partial p_k} \right|_{\boldsymbol{p}^*} \right) \tag{1} \end{equation}\]Write the expression for each derivative individually, replacing the last variable: \(p_m = 1 - \sum_{j=1}^{m-1} p_j\).
To get the left-hand term in the Jacobian element, start with
\[\begin{align} f_i &= \sum_{j=1}^m p_j \pi(i \mid j) \\ &= \left[ \sum_{j=1}^{m-1} p_j \pi(i \mid j) \right] + \pi(i \mid m) \left[ 1 - \sum_{j=1}^{m-1} p_j \right] \\ &=\left[ \sum_{j=1}^{m-1} p_j [\pi(i \mid j) - \pi(i \mid m)] \right] + \pi(i \mid m) \end{align}\]Therefore:
\[\begin{equation} \left. \frac{\partial f_i}{\partial p_k} \right|_{\boldsymbol{p}^*} = \pi(i \mid k) - \pi(i \mid m) \tag{left-hand term of (1)} \end{equation}\]For the right-hand term in the Jacobian element, split the population average fitness effect into three terms
\[\begin{equation} \overline{f} = \sum_j p_j f_j = \left[ \sum_{j=1, j\neq k}^{m-1} p_j f_j \right] + p_k f_k + p_m f_m \end{equation}\]and take the derivatives of each term separately.
The first term:
\[\begin{align} \left. \frac{\partial}{\partial p_k} \left[ \sum_{j=1, j \neq k}^{m-1} p_j f_j \right] \right|_{\boldsymbol{p}^*} &= \sum_{j=1, j \neq k}^{m-1} p_j^* \left. \frac{\partial f_j}{\partial p_k} \right|_{\boldsymbol{p}^*} \\ &=\sum_{j=1, j \neq k}^{m-1} p_j^* [\pi(j \mid k) - \pi(j \mid m)] \end{align}\]The second term:
\[\begin{align} \left. \frac{\partial \left[ p_k f_k \right]}{\partial p_k} \right|_{\boldsymbol{p}^*} &= \left. f_k \right|_{\boldsymbol{p}^*} + p_k^* \left. \frac{\partial f_k}{\partial p_k} \right|_{\boldsymbol{p}^*} \\ &= \left[ \sum_{j=1}^m p^*_j \pi(k \mid j) \right] + p_k^* [\pi(k \mid k) - \pi(k \mid m)] \end{align}\]The third term:
\[\begin{align} \left. \frac{\partial \left[ p_m f_m \right]}{\partial p_k} \right|_{\boldsymbol{p}^*} &= \left. \frac{\partial }{\partial p_k} \left[1-\sum_{j=1}^{m-1} p_j \right] \right|_{\boldsymbol{p}^*} \left. f_m \right|_{\boldsymbol{p}^*} + p_m^* \left. \frac{\partial f_m}{\partial p_k} \right|_{\boldsymbol{p}^*} \\ &= - \left[ \sum_{j=1}^m p_j^* \pi(m \mid j) \right] + p_m^* [\pi(m \mid k) - \pi(m \mid m)] \end{align}\]So summing them together to get the right-hand term in the Jacobian element:
\[\begin{equation} \left. \frac{\partial \overline{f}}{\partial p_k} \right|_{\boldsymbol{p}^*} = \sum_{j=1}^m p_j^* [ \pi(j \mid k) - \pi(j \mid m) + \pi(k \mid j) - \pi(m \mid j) ] \tag{right-hand term of (1)} \end{equation}\]Let’s now code the Jacobian.
First, code the payoffs so \(\pi(j \mid k) =\) pays[j][k]
and the interior steady state \(p_j^* =\) ps[j]
.
Code the right-hand term of (1), \(\begin{equation} \left. \frac{\partial \overline{f}}{\partial p_k} \right|_{\boldsymbol{p}^*} \end{equation}\)
Code the full \(J_{i,k}\) in (1)
We find that the Jacobian matrix is
and its eigenvalues
The maximum real part is positive, so the interior steady state is unstable.
Ohtsuki, H., & Nowak, M. A. (2006). The replicator equation on graphs. Journal of Theoretical Biology, 243(1), 86-97.
]]>The paper gives many detailed examples and arguments for why real-world social kinship has the features we would expect if group nepotism — combining relatedness with collective action — was an important mechanism (and also a section on ethnocentrism), which I cannot do justice here. Instead, the purpose of this blog post is to simply to teach myself about the basic idea by working through the first model in the paper.
The first model is called the Brothers Karamazov Game, named after the Dostoevsky novel. I’ve never read The Brothers Karamazov, but I gather there are three brothers — Ivan, Alyosha, and Dmitri — and Dmitri is prone to making poor life choices. The question is, should Ivan and Alyosha help out their brother Dmitri?
When Ivan and Alyosha act independently, then the rule for helping Dmitri to a benefit \(B\) at a cost \(C\) follows directly from Hamilton’s rule. The coefficient of relatedness between full siblings is \(r=1/2\); therefore, Ivan (or Alyosha) should help Dmitri if the benefit to Dmitri is at least twice the cost, \(B/C > 1/r = 2\). However, Jones shows this constraint can be made much easier to overcome if Ivan and Alyosha act together.
Jones introduces a situation he calls “conditional nepotism”: Ivan proposes to Alyosha that he will help out Dmitri if and only if Alyosha agrees to do the same. If Alyosha agrees to this, then when we add up Ivan’s inclusive fitness costs and benefits (details below), the pooling of nepotistic effort leads for a condition for helping \(B/C > 1/r_c\) where \(r_c > r\). In other words, provided Alyosha agrees, then Ivan should treat Dmitri as though he is even more related to him than he really is. This is an interesting result because it might help explain why helping between kin — culturally defined — is common in hunter-gatherer societies even though relatedness coefficients between pairs of individuals are often quite low.
I will now work through the sexual haploid case in the first appendix. However, instead of using Jones’ method, I will try performing the calculations using the framework provided in Ohtsuki (2014; Proc R Soc B).
Ohtsuki’s method allows us to describe the replicator dynamics of two strategies \(A\) and \(B\) in nonlinear games played between kin by taking into account the higher-order genetic associations between group members, beyond their dyadic relatedness. Dyadic relatedness \(r\) above is the probability that 2 individuals randomly drawn without replacement from the group will have strategies that are identical by descent, denoted \(\theta_{2 \rightarrow 1}\) in Ohtsuki’s scheme. We can also say in shorthand that the two individuals “share a common ancestor”. Higher-order relatedness generalises this concept to larger samples. For example, \(\theta_{3 \rightarrow 1}\) is the probability that three individuals randomly drawn share a common ancestor, and in general, \(\theta_{l \rightarrow m}\) is the probability that, if we draw \(l\) individuals from the group, they share \(m\) common ancestors.
The higher-order genetic associations \(\theta_{l \rightarrow m}\) can then be used to find the probilities \(\rho_l\) that \(l\) players randomly sampled from the group without replacement are \(A\)-strategists (Eq. 2.5):
\[\rho_l = \sum_{m=1}^l \theta_{l \rightarrow m} p^m\]where \(p\) is the proportion of \(A\)-strategists in the population and \(\rho_0=1\).
The reasoning behind the \(\rho_l\) equation can be seen by considering the probability that two randomly sampled group members are \(A\)-strategists, \(\rho_2\). Either the two individuals have the same ancestor, or they have different ancestors. Assuming that the proportion of \(A\)-strategists in the ancestral population can be approximated by the proportion of \(A\)-strategists now, then:
\[\rho_2 = \quad \underbrace{\theta_{2 \rightarrow 1}}_{\substack{\text{same} \\ \text{ancestor}}} \quad \underbrace{p}_{\substack{\text{ancestor} \\ \text{=A}} } \quad + \quad \underbrace{\theta_{2 \rightarrow 2}}_{\substack{\text{two} \\ \text{ancestors}} } \quad \underbrace{p^2}_{\substack{\text{both} \\ \text{=A}}}\]Let’s define the payoffs to the two strategies using two functions: \(a_k\) is the payoff to \(A\)-strategists when \(k\) of the \(n-1\) other group members are \(A\)-strategists, and \(b_k\) is the payoff to \(B\)-strategists when \(k\) of the \(n-1\) other group members are also \(A\)-strategists.
Then Ohtsuki (2014) shows that the change in the proportion \(A\)-strategists in the population, \(p\), over one generation is proportional to (Eq. 2.4)
\[\Delta p \propto \sum_{k=0}^{n-1} \sum_{l=k}^{n-1} (-1)^{l-k} {l \choose k} {n-1 \choose l} \left[ (1-\rho_1) \rho_{l+1} a_k - \rho_1(\rho_l - \rho_{l+1}) b_k \right].\]This equation is derived from first principles from the Price equation (see Ohtsuki (2014) appendices).
The number of terms in the dynamics equation quickly becomes unweildy, so I’ll use SageMath below to work through the algebra.
First, I will prepare the symbolic function “\(\Delta p \propto \ldots\)”
\[\begin{gather} -{\left(a\left(0\right) + 2 \, b\left(1\right) - 3 \, b\left(0\right)\right)} \rho\left(1\right)^{2} - {\left({\left(\rho\left(1\right) - 1\right)} a\left(2\right) - {\left(2 \, a\left(1\right) - a\left(0\right) - 2 \, b\left(1\right) + b\left(0\right)\right)} \rho\left(1\right) - b\left(2\right) \rho\left(1\right) + 2 \, a\left(1\right) - a\left(0\right)\right)} \rho\left(3\right) \\ - {\left({\left(2 \, a\left(1\right) - 2 \, a\left(0\right) - 4 \, b\left(1\right) + 3 \, b\left(0\right)\right)} \rho\left(1\right) + b\left(2\right) \rho\left(1\right) - 2 \, a\left(1\right) + 2 \, a\left(0\right)\right)} \rho\left(2\right) - {\left(b\left(0\right) \rho\left(0\right) - a\left(0\right)\right)} \rho\left(1\right) \end{gather}\]The example in Section 4(b) of Ohtsuki (2014) provides us with expressions for the higher-order relatedness terms between siblings. Define the dyadic relatedness \(r = \theta_{2 \rightarrow 1}\) as before, and define the triplet relatedness \(s = \theta_{3 \rightarrow 1}\). Then it can be shown:
\[\begin{align} \theta_{1 \rightarrow 1} &= 1, \\ \theta_{2 \rightarrow 1} &= r, \\ \theta_{2 \rightarrow 2} &= 1-r, \\ \theta_{3 \rightarrow 1} &= s, \\ \theta_{3 \rightarrow 2} &= 3r - 3s, \\ \theta_{3 \rightarrow 3} &= 1 - 3r + 2s. \end{align}\]Between siblings, \(r = 1/2\) and \(s = 1/4\), therefore
\[\begin{align} \rho_1 &= \theta_{1 \rightarrow 1} p = p, \\ \rho_2 &= \theta_{2 \rightarrow 1} p + \theta_{2 \rightarrow 2} p^2 = \frac{p}{2} + \frac{p^2}{2}, \\ \rho_3 &= \theta_{3 \rightarrow 1} p + \theta_{3 \rightarrow 2} p^2 + \theta_{3 \rightarrow 3} p^3 = \frac{p}{4} + \frac{3p^2}{4} \end{align}\]We will substitute these expressions for \(\rho_l\) into the symbolic function:
\[\begin{gather} -p^{2} {\left(a\left(0\right) + 2 \, b\left(1\right) - 3 \, b\left(0\right)\right)} + \frac{1}{4} \, {\left(3 \, p^{2} + p\right)} {\left(p {\left(2 \, a\left(1\right) - a\left(0\right) - 2 \, b\left(1\right) + b\left(0\right)\right)} - {\left(p - 1\right)} a\left(2\right) + p b\left(2\right) - 2 \, a\left(1\right) + a\left(0\right)\right)} \\ - \frac{1}{2} \, {\left(p^{2} + p\right)} {\left(p {\left(2 \, a\left(1\right) - 2 \, a\left(0\right) - 4 \, b\left(1\right) + 3 \, b\left(0\right)\right)} + p b\left(2\right) - 2 \, a\left(1\right) + 2 \, a\left(0\right)\right)} + p {\left(a\left(0\right) - b\left(0\right)\right)} \end{gather}\]Now we have an expression proportional to \(\Delta p\) to which we can apply different payoff functions \(a_k\) and \(b_k\).
First, let’s consider the case where the two brothers always help Dmitri regardless of what the other does. We already know the answer whether or not this behaviour can evolve, it follows directly from Hamilton’s Rule, so this also serves to sanity-check the calculations.
Let the \(A\) strategy be unconditional helping and \(B\) be defection. When the focal player is playing the role of Dmitri, which is drawn with probability \(1/3\), their payoffs are independent of their strategy
\[\begin{pmatrix} a_0 & a_1 & a_2 \\ b_0 & b_1 & b_2 \end{pmatrix} = \begin{pmatrix} 0 & B & 2B \\ 0 & B & 2B \end{pmatrix}\]If the focal is not a Dmitri (occurs with probability 2/3), the payoffs to the focal depend on whether or not they help the Dmitri
\[\begin{pmatrix} a_0 & a_1 & a_2 \\ b_0 & b_1 & b_2 \end{pmatrix} = \begin{pmatrix} -C & -C & -C \\ 0 & 0 & 0 \end{pmatrix}\]Therefore, when the brothers act independently, the expected payoffs
\[\begin{pmatrix} a_0 & a_1 & a_2 \\ b_0 & b_1 & b_2 \end{pmatrix} = \frac{1}{3} \begin{pmatrix} -2C & B-2C & 2B-2C \\ 0 & B & 2B \end{pmatrix}\]Substituting these payoffs into the expression proportional to \(\Delta p\)
\[-\frac{1}{3} \, {\left(B - 2 \, C\right)} {\left(p - 1\right)} p\]The condition for the \(A\)-strategist (the ones who help Dmitri) to increase is
\[B-2C > 0\]and the condition is
\[B/C > 2 = 1/r\]as expected.
Now let’s consider Jones’ scenario, where Ivan and Alyosha help Dmitri if and only if they both agree, i.e., they are both conditional nepotists.
You might notice in the first appendix that Jones talks about a ‘conditional nepotists’ vs ‘Hamiltonian nepotists’ scenario, which confused me at first, because when you look at the payoffs Table A2, there’s no helping by the Hamiltonian \(H\) type. I think what’s happening is he is actually restricting his attention to the situation \(B/C < 2\), and assuming that what he calls ‘Hamiltonian nepotists’ will decide to not help Dmitri in this situation, i.e., act like defectors (see wording of the paragraph right after Eq. A1). Therefore, the scenario in question is actually ‘conditional nepotists vs defectors’. Hopefully this will become clear by the time we’ve gone through all the cases (final figure summary).
Let the \(A\) strategy be conditional nepotism and the \(B\) strategy be defection. When the focal player is a Dmitri (probability \(1/3\)), their payoffs are again independent of their strategy
\[\begin{pmatrix} a_0 & a_1 & a_2 \\ b_0 & b_1 & b_2 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 2B \\ 0 & 0 & 2B \end{pmatrix}\]When the focal player is not a Dmitri (probability 2/3), their payoffs are
\[\begin{pmatrix} a_0 & a_1 & a_2 \\ b_0 & b_1 & b_2 \end{pmatrix} = \begin{pmatrix} 0 & -C/2 & -C \\ 0 & 0 & 0 \end{pmatrix}\]where \(a_1 = -C/2\) because, if there is one other helper in the group, there is half a chance that the other helper is also not a Dmitri and thus the \(A\)-strategist gives the help.
Therefore, when the brothers act as conditional nepotists, the expected payoffs
\[\begin{pmatrix} a_0 & a_1 & a_2 \\ b_0 & b_1 & b_2 \end{pmatrix} = \frac{1}{3} \begin{pmatrix} 0 & -C & 2B-2C \\ 0 & 0 & 2B \end{pmatrix}\] \[-\frac{1}{6} \, {\left(2 \, B p - 2 \, C p + B - 2 \, C\right)} {\left(p - 1\right)} p\]The condition for conditional nepotists to increase matches the equation that Jones presents in the appendix (Eq. A1)
\[2Bp - 2Cp + B - 2C > 0\]or
\[\frac{B}{C} > \frac{2 + 2p}{1 + 2p} \equiv \frac{1}{r_c}.\]As \(p\) increases from 0 to 1, \(r_c\) above increases from 1/2 to 3/4, meaning that when \(p\) is high, the brothers will treat Dmitri as though he is more related to them than he really is. Conditional nepotists can invade defectors when (sub in \(p=0\)) \(B/C > 2\). Defectors can invade conditional nepotists (sub in \(p=1\), reverse condition) when \(B/C < 4/3\).
Let’s plot some examples of the \(\Delta p\) vs \(p\) function to get a more intuitive feel. When \(B/C = 2.5 > 2\), we see below that \(\Delta p\) is always greater than zero.
When \(B/C = 1.5\) (satisfying \(4/3 < B/C < 2\)), we see that \(\Delta p\) is less than zero when \(p\) is small, and greater than zero when \(p\) is large. There are two stable steady states — all Defectors and all Conditional Nepotists — and a separatrix between them. The position of the separatrix is found by setting the \(B/C\) condition above to an equality: \(p^{\star} = \frac{2C-B}{2B-2C}\).
When \(B/C = 1.2 < 4/3\), \(\Delta p\) is always below zero. All-defectors is the evolutionary endpoint.
For completeness, we should now consider the case of ‘conditional nepotists’ vs ‘unconditional helpers’. Let the \(A\) strategy be conditional nepotism and the \(B\) strategy be unconditional helping. Substituting in the expected payoffs:
\[-\frac{1}{6} \, {\left(2 \, B p - 2 \, C p - B + 2 \, C\right)} {\left(p - 1\right)} p\]The condition for conditional nepotists to increase is
\[B(2p-1) > C(2p-2)\]which splits up into two cases. When \(p > 1/2\), the condition becomes
\[\frac{B}{C} > \frac{2p-2}{2p-1}\]and so conditional nepotists can always grow. When \(p < 1/2\), the condition becomes
\[\frac{B}{C} < \frac{2p-2}{2p-1}\]and conditional nepotists can only invade (\(p=0\)) if \(B/C < 2\).
Let’s again plot some examples to obtain the intuition. First, let’s plot an example where \(B/C < 2\). The \(p^*=0\) steady state is unstable, \(p^* =1\) is stable, and there are no interior equilibria. Therefore, conditional nepotists can invade, and an all-conditional-nepotists population is the evolutionary endpoint.
Now let’s see how the situation changes as we move to the \(B/C > 2\) space. In the plot below, we see how, when we reach \(B/C = 2\), an interior separatrix appears, so now there are two evolutionarily stable states: all conditional nepotists, and all unconditional helpers. So conditional nepotists cannot invade, but once established, they can resist invasion by unconditional helpers.
As \(B/C\) becomes larger, the separatrix moves to the right. When we plot an example with large \(B/C\), the separatrix is close to \(p \approx 1/2\), as suggested above.
All evolutionary endpoints are monomorphic, so I summarised the evolutionary dynamics in terms of pairwise invasibility below.
When benefits are high compared to costs, defectors cannot prevail. The evolutionary endpoints are conditional nepotists or unconditional helpers. When benefits are intermediate \(4/3 < B/C < 2\), unconditional helpers are no longer evolutionarily stable. Either defectors or conditional nepotists will prevail. When benefits are low, defectors prevail.
Conditional nepotists can never invade. However, if the benefits are close to but just above \(B/C = 2\), the separatrix between conditional nepotists and unconditional helpers is close to zero. This suggests that, if a cluster of unconditional helpers have a brainwave and strike upon the idea of making a deal like Jones describes, then they might be able to invade and explore the new space of potential collaborative situations not previously accessible to the unconditional helpers. The separatrix between conditional nepotists and defectors is near 0 when \(B/C\) is near \(2\), so they will (at least, initially), be quite resistant to invasion by defectors.
The Brothers Karamazov model above is only the first of 3 models that Jones (2000) discusses. In the second model, Jones considers a large donor group facing the choice of whether to help another group of relatives. Through a similar principle to “conditional nepotism” above, the donor group decides as a collective whether or not to help, i.e., “group nepotism”. Analogous to \(r_c\) above, “group coefficient of relatedness” \(r_g\) is obtained from the condition \(B/C > 1/r_g\), and it too can be larger than dyadic relatedness \(r\).
Table 1 below compares estimates of the dyadic relatedness (labelled \(r_{11}\) in the talbe) and the group coefficient of relatedness for a sampling of tribal societies. Group relatedness is often much higher than dyadic relatedness, sometimes nearing 1. This suggests that, if groups of relatives can somehow coordinate themselves and act as a collective, then they will render help more easily than one would expect just from individual inclusive fitness considerations alone.
Jones proposes that such collective behaviour might be achieved by “mutual coercion, mutually agreed upon”, and gives many real-life anthropological examples where individuals are punished for not obeying norms about how one ought to treat their (culturally defined) kin. Models that also include coercion (punishment) could be used to learn more about the exact conditions under which this can evolve. So I have more reading to do.
Jones, D. (15 June, 2016). Blog post: “Beating Hamilton’s Rule”, https://logarithmichistory.wordpress.com/2016/06/15/beating-hamiltons-rule/
Jones, D. (2000). Group nepotism and human kinship. Current Anthropology, 41(5): 779-809
Ohtsuki, H. (2014). Evolutionary dynamics of n-player games played by relatives. Philosophical Transactions of the Royal Society B: Biological Sciences, 369(1642), 20130359.
]]>De Jaegher used the replicator dynamics approach. Replicator dynamics assumes a well-mixed, infinitely large population of players that reproduces asexually, where the number of offspring they produce depends upon the payoff they receive from a game-theoretic ‘game’, as illustrated below:–
We start with a population of \(n\) individuals with different game strategies. Individuals are randomly selected to form groups and play the game. The payoff from the game determines how many offspring they have. Offspring inherit the strategies of their parents (clonal reproduction), and the cycle repeats.
Let’s rederive the dynamics. The number of individuals pursuing strategy \(i\) changes according to
\[\frac{dn_i}{dt} = \dot{n_i} = n_i (\beta + f_i),\]where \(\beta\) is the background reproduction rate (apart from the game’s effect), and \(f_i\) is the fitness effect of playing strategy \(i\).
Denote the proportion of \(i\)-strategists in the population
\[p_i = \frac{n_i}{N}\]where \(N\) is the total population size.
We want to know \(\frac{dp_i}{dt}\). Rearrange the \(p_i\) definition and take the derivatives
\[n_i = p_i N\] \[\dot{n_i} = p_i \dot{N} + \dot{p_i} N\] \[\dot{p} = \frac{1}{N} (\dot{n_i} - p_i \dot{N})\]So we need to sort out \(\dot{N}\)
\[\begin{align} \dot{N} &= \sum_i \dot{n_i} \\ &= \beta \sum_i n_i + \sum_i f_i n_i \\ &= \beta N + N \sum_i f_i \frac{n_i}{N} \\ &= \beta N + N \sum_i f_i p_i \\ \dot{N} &= N (\beta +\bar{f}) \\ \end{align}\]Substitute our equations for \(\dot{N}\) and \(\dot{n_i}\) into the equation for \(\dot{p_i}\), and we obtain the dynamics of strategy proportions
\[\frac{dp_i}{dt} = \dot{p} = p_i (f_i - \bar{f})\]In the De Jaegher paper, players face the binary choice of cooperating or defecting. Denote cooperators and defectors by C and D, respectively, and define \(p = p_C = 1-p_D\). Then the equation governing the dynamics simplifies to
\[\begin{align} \dot{p} &= p \, (f_C - [p f_C + (1-p) f_D]) \\ \dot{p} &= p (1-p) (f_C - f_D) \end{align}\]In the threshold game, a group is formed with \(n\) players (different to \(n\)) above, and if the group contains \(k\) or more cooperators, then they provide a benefit \(b\). The benefit \(b\) is received by all group members regardless of whether that member was a cooperator or defector
\[b_i = \begin{cases} b &\text{if } i >= k \text{ (threshold met)}\\ 0 &\text{otherwise (threshold not met)} \end{cases}\]Throughout their paper, they have normalised \(b=1\).
Everyone receives the benefit, but only cooperators pay the cost \(c\). So the total payoff to cooperators is
\[\text{cooperator payoff} = \begin{cases} b - c & \text{if threshold met,} \\ -c & \text{if threshold not met.} \end{cases}\]and defectors receive
\[\text{defector payoff} = \begin{cases} b & \text{if threshold met,} \\ 0 & \text{if threshold not met.} \end{cases}\]Defectors always do better than cooperators.
The payoffs above tell us about what happens in a particular game with a particular number of cooperators and defectors, but we need the fitness effects of being a cooperator and defector, \(f_C\) and \(f_D\), which are averaged over all the games played.
The replicator dynamics assumes that groups are formed randomly from an infinite population, so the make-up of the other players is binomially distributed. Therefore, the fitness effect is the sum of the payoffs from each game multiplied by the probability that the player will end up in that game
\[f_C(p) = \sum_{i=0}^{n-1} \underbrace{ {n-1 \choose i} p^i (1-p)^{n-1-i}}_{\text{Pr grouped with $i$ cooperators}} \: b_{i+1} - c,\] \[f_D(p) = \sum_{i=0}^{n-1} \underbrace{ {n-1 \choose i} p^i (1-p)^{n-1-i}}_{\text{Pr grouped with $i$ cooperators}} \: b_{i}.\]The pivot probability is the probability that a player will end up in a game where \(k-1\) of the other \(n-1\) players is a cooperator
\[\pi_k = {n-1 \choose k-1} p^{k-1} (1-p)^{n-k}.\]Why is the pivot probability significant? First, consider just a single game, and imagine I know that I am the pivotal player. If I am a cooperator, then it is in my interests to remain a cooperator because if I switch then I will reduce my own payoff. If I am defector and I have the option of switching strategies, I would want to switch to cooperate.
We can also intuit that, if the probability that I will end up in a game where I am the pivotal player is high, then assuming I have a fixed strategy that I play all the time, it might be in my interests to be a cooperator… We’ll return to this point soon.
Let’s just plot an example to start with to get a sense of the dynamics. Recall
\[\dot{p} = p (1-p) (f_C - f_D)\]where \(f_C\) and \(f_D\) defined in the paper. Let us plot this.
In the figure below, we find that the all-defector population is an evolutionarily stable steady state, and the all-cooperation population is unstable. There is also an evolutionarily stable state with a mix of cooperators and defectors (at \(p^{\star} = 0.746\)).
We can also see some of this directly by inspecting the equation for the dynamics. Recall
\[\frac{dp}{dt} = p (1-p) (f_C(p) - f_D(p)).\]Steady states occur when \(\frac{dp}{dt} =0\), so there are two trivial steady states
\[p = 0,\]and
\[p = 1.\]and any interior steady states solve
\[0 = f_C(p) - f_D(p).\]Let’s focus on these interior steady states. Recall
\[f_C(p) = \sum_{i=0}^{n-1} { n-1 \choose i } p^i (1-p)^{n-1-i} b_{i+1} - c,\]and
\[f_D(p) = \sum_{i=0}^{n-1} { n-1 \choose i } p^i (1-p)^{n-1-i} b_i,\]but the \(b_i\) values are only 1 when \(i >=k\), otherwise 0. Therefore, we can simplify
\[f_C(p) - f_D(p) = \underbrace{ {n-1 \choose k-1} p^{k-1} (1-p)^{n-k}}_{\pi_k(p)} - c,\]and notice that the pivot probability is in it.
In summary, the dynamics are
\[\frac{dp}{dt} = p \, (1-p) \, \overbrace{ \underbrace{ {n-1 \choose k-1} p^{k-1} (1-p)^{n-k} }_{\pi_k(p)} - c}^{f_C(p) - f_D(p)}\]Let’s plot just the function that solves the interior steady states
\[\pi_k(p) - c = 0.\]De Jaegher found that, when \(1 < k < n\), both \(f_C - f_D\) and \(\pi_k\) have a peak at \(\hat{p} = \frac{k-1}{n-1}\). They found this by taking advantage of the fact that the function is unimodal, and finding when the derivative was 0.
\[f_C(p) - f_D(p) = {n-1 \choose k-1} p^{k-1} (1-p)^{n-k} - c,\]so
\[\frac{d(f_C(p) - f_D(p))}{dp} = {n-1 \choose k-1} (1-p)^{n-k-1} p^{k-2} \bigl( (k-1)(1-p) + (k-n) p \bigr).\]The peak is located where \(\frac{d(f_C(p) - f_D(p))}{dp} = 0\) that is
\[0 = (k-1)(1-\hat{p}) + (k-n) \hat{p},\]which gives the peak location
\[\hat{p} = \frac{k-1}{n-1}.\]Because \(f_C - f_D\) and \(\pi_k\) differ only by a constant, this point is also the location of the peak of the pivot probability function.
Knowing where this peak is and whether it is above or below the line tells us about the dynamics, as will become clearer in the examples below.
Below, we’ll go through examples for each of the cases in Result 1.
For the minimal threshold \(k = 1\), the game has a unique interior fxed point where a fraction \(p_1^{\text{II}} = 1-c^{1/(n-1}\) of players cooperates (Volunteer’s Dilemma).
Low cost
When participation costs are small (\(c < \bar{c}_k\)), the game both has a fixed point where all players defect (\(p = 0\)) and a stable fixed point \(p_k^{\text{II}}\), which is implicitly given by \(\pi_k(p_k^{\text{II}}) = c\).
High cost
When participation costs are high, \(c > \bar{c}_k\), the game has a unique stable fixed point where all players defect (\(p = 0\)).
What’s the significance of this \(\bar{c}_k\) value? Recall
\[f_C(p) - f_D(p) = \underbrace{ {n-1 \choose k-1} p^{k-1} (1-p)^{n-k}}_{\pi_k(p)} - c.\]They’ve defined
\[\bar{c}_k = \pi_k(\hat{p}),\]where \(\hat{p}\) is the probability at the peak. So we know that, if \(\bar{c}_k > c\), the \(f_C - f_D\) line will be above the zero line at the peak and there will be an interior equilibrium, and if \(\bar{c}_k < c\), the peak is below the line and there are no interior equilibria.
Our final case is the maximum threshold, \(k=n\). Here, the game both has a stable steady state where all players defect (\(p = 0\)), and a stable steady state where all players cooperate (\(p = 1\)). The basin of attraction is defined by \(p_n^{\text{I}} = c^{1/(n-1)}\), the unstable interior steady state.
We can place each of these examples on De Jaegher’s Fig. 4. The 4 regions of the figure represent four different qualitative regimes for the dynamics (game types).
De Jaegher found that the threshold level has a U-shaped efect on the level of cooperation, which can be seen in their Fig. 4 (the shape of the division between the blue and green region). Let’s plot how the dynamics varies with varying threshold level.
Here, we see that cooperation can evolve both for low and for high thresholds, but not for intermediate thresholds. It is perhaps surprising because, intuitively, it seems like a high threshold would be more difficult to obtain.
This U-shape emerges from the pivot probabilities.
De Jaegher discuss the possibility of stabilising a game by changing the group size. If cooperation is more likely to persist at low or high thresholds, then perhaps increasing or decreasing the group size to shift the relative position of the threshold could stabilise cooperation.
It turns out this is true for decreasing the group size, but not for increasing it. Increasing the group size has a negative effect on cooperation.
The reason why is due to the effect of group size on the pivot probability: the larger the group is, the less likely it is for a player to be the pivotal player, and so the lower the benefit of being a cooperator.
De Jaegher, K. (2020). High thresholds encouraging the evolution of cooperation in threshold public-good games. Scientific Reports, 10(1), 1-10.
]]>