Heinrich H. Nax

Spread of yellow fever virus outbreak in Angola and the Democratic Republic of the Congo 2015–16: a modelling study

Summary Background Since late 2015, an epidemic of yellow fever has caused more than 7334 suspected cases in Angola and the Democratic Republic of the Congo, including 393 deaths. We sought to understand the spatial spread of this outbreak to optimise the use of the limited available vaccine stock. Methods We jointly analysed datasets describing the epidemic of yellow fever, vector suitability, human demography, and mobility in central Africa to understand and predict the spread of yellow fever virus. We used a standard logistic model to infer the district-specific yellow fever virus infection risk during the course of the epidemic in the region. Findings The early spread of yellow fever virus was characterised by fast exponential growth (doubling time of 5–7 days) and fast spatial expansion (49 districts reported cases after only 3 months) from Luanda, the capital of Angola. Early invasion was positively correlated with high population density (Pearsons r 0·52, 95% CI 0·34–0·66). The further away locations were from Luanda, the later the date of invasion (Pearsons r 0·60, 95% CI 0·52–0·66). In a Cox model, we noted that districts with higher population densities also had higher risks of sustained transmission (the hazard ratio for cases ceasing was 0·74, 95% CI 0·13–0·92 per log-unit increase in the population size of a district). A model that captured human mobility and vector suitability successfully discriminated districts with high risk of invasion from others with a lower risk (area under the curve 0·94, 95% CI 0·92–0·97). If at the start of the epidemic, sufficient vaccines had been available to target 50 out of 313 districts in the area, our model would have correctly identified 27 (84%) of the 32 districts that were eventually affected. Interpretation Our findings show the contributions of ecological and demographic factors to the ongoing spread of the yellow fever outbreak and provide estimates of the areas that could be prioritised for vaccination, although other constraints such as vaccine supply and delivery need to be accounted for before such insights can be translated into policy. Funding Wellcome Trust.

Directional learning and the provisioning of public goods.

We consider an environment where players are involved in a public goods game and must decide repeatedly whether to make an individual contribution or not. However, players lack strategically relevant information about the game and about the other players in the population. The resulting behavior of players is completely uncoupled from such information, and the individual strategy adjustment dynamics are driven only by reinforcement feedbacks from each players own past. We show that the resulting “directional learning” is sufficient to explain cooperative deviations away from the Nash equilibrium. We introduce the concept of k–strong equilibria, which nest both the Nash equilibrium and the Aumann-strong equilibrium as two special cases, and we show that, together with the parameters of the learning model, the maximal k–strength of equilibrium determines the stationary distribution. The provisioning of public goods can be secured even under adverse conditions, as long as players are sufficiently responsive to the changes in their own payoffs and adjust their actions accordingly. Substantial levels of public cooperation can thus be explained without arguments involving selflessness or social preferences, solely on the basis of uncoordinated directional (mis)learning.

Stability of cooperation under image scoring in group interactions

Image scoring sustains cooperation in the repeated two-player prisoner’s dilemma through indirect reciprocity, even though defection is the uniquely dominant selfish behaviour in the one-shot game. Many real-world dilemma situations, however, firstly, take place in groups and, secondly, lack the necessary transparency to inform subjects reliably of others’ individual past actions. Instead, there is revelation of information regarding groups, which allows for ‘group scoring’ but not for image scoring. Here, we study how sensitive the positive results related to image scoring are to information based on group scoring. We combine analytic results and computer simulations to specify the conditions for the emergence of cooperation. We show that under pure group scoring, that is, under the complete absence of image-scoring information, cooperation is unsustainable. Away from this extreme case, however, the necessary degree of image scoring relative to group scoring depends on the population size and is generally very small. We thus conclude that the positive results based on image scoring apply to a much broader range of informational settings that are relevant in the real world than previously assumed.

Payoff-based learning explains the decline in cooperation in public goods games.

Economic games such as the public goods game are increasingly being used to measure social behaviours in humans and non-human primates. The results of such games have been used to argue that people are pro-social, and that humans are uniquely altruistic, willingly sacrificing their own welfare in order to benefit others. However, an alternative explanation for the empirical observations is that individuals are mistaken, but learn, during the game, how to improve their personal payoff. We test between these competing hypotheses, by comparing the explanatory power of different behavioural rules, in public goods games, where individuals are given different amounts of information. We find: (i) that individual behaviour is best explained by a learning rule that is trying to maximize personal income; (ii) that conditional cooperation disappears when the consequences of cooperation are made clearer; and (iii) that social preferences, if they exist, are more anti-social than pro-social.

A behavioral study of “noise” in coordination games

‘Noise’ in this study, in the sense of evolutionary game theory, refers to deviations from prevailing behavioral rules. Analyzing data from a laboratory experiment on coordination in networks, we tested ‘what kind of noise’ is supported by behavioral evidence. This empirical analysis complements a growing theoretical literature on ‘how noise matters’ for equilibrium selection. We find that the vast majority of decisions (96%96%) constitute myopic best responses, but deviations continue to occur with probabilities that are sensitive to their costs, that is, less frequent when implying larger payoff losses relative to the myopic best response. In addition, deviation rates vary with patterns of realized payoffs that are related to trial-and-error behavior. While there is little evidence that deviations are clustered in time or space, there is evidence of individual heterogeneity.

Evolutionary dynamics and equitable core selection in assignment games

We study evolutionary dynamics in assignment games where many agents interact anonymously at virtually no cost. The process is decentralized, very little information is available and trade takes place at many different prices simultaneously. We propose a completely uncoupled learning process that selects a subset of the core of the game with a natural equity interpretation. This happens even though agents have no knowledge of other agents’ strategies, payoffs, or the structure of the game, and there is no central authority with such knowledge either. In our model, agents randomly encounter other agents, make bids and offers for potential partnerships and match if the partnerships are profitable. Equity is favored by our dynamics because it is more stable, not because of any ex ante fairness criterion.

Assortativity Evolving from Social Dilemmas

Assortative mechanisms can overcome tragedies of the commons that otherwise result in dilemma situations. Assortativity criteria include various forms of kin selection, greenbeard genes, and reciprocal behaviors, usually presuming an exogenously fixed matching mechanism. Here, we endogenize the matching process with the aim of investigating how assortativity itself, jointly with cooperation, is driven by evolution. Our main finding is that full-or-null assortativities turn out to be long-run stable in most cases, independent of the relative speeds of both processes. The exact incentive structure of the underlying social dilemma matters crucially. The resulting social loss is evaluated for general classes of dilemma games, thus quantifying to what extent the tragedy of the commons may be endogenously overcome.

A Note on the Core of TU-cooperative Games with Multiple Membership Externalities

A generalization of transferable utility cooperative games from the functional forms introduced by von Neumann and Morgenstern (1944, Theory of Games and Economic Behavior) and Lucas and Thrall (1963, Naval Research Logistics Quarterly, 10, 281–298) is proposed to allow for multiple membership. The definition of the core is adapted analogously and the possibilities for the cross-cutting of contractual arrangements are illustrated and discussed.

Core Stability and Core Selection in a Decentralized Labor Matching Market

We propose a dynamic model of decentralized many-to-one matching in the context of a competitive labor market. Through wage offers and wage demands, firms compete over workers and workers compete over jobs. Firms make hire-and-fire decisions dependent on the wages of their own workers and on the alternative workers available on the job market. Workers bargain for better jobs; either individually or collectively as unions, adjusting wage demands upward/downward depending on whether they are currently employed/unemployed. We show that such a process is absorbed into the core with probability one in finite time. Moreover, within the core, allocations are selected that are characterized by surplus splitting according to a bargaining solution such that (i) firms and workforce share total revenue according to relative bargaining strengths, and (ii) workers receive equal workforce shares above their individual outside options. These results bridge empirical evidence and provide a rich set of testable predictions.

Meritocratic matching can dissolve the efficiency-equality tradeoff: the case of voluntary contributions

Real-world institutions dealing with social dilemma situations are based on mechanisms that are rarely implemented without flaw. Usually real-world mechanisms are noisy and imprecise, that is, which we call ‘fuzzy’. We therefore conducted a novel type of voluntary contributions experiment where we test a mechanism by varying its fuzziness. We focus on a range of fuzzy mechanisms we call ‘meritocratic matching’. These mechanisms generalize the mechanism of ‘contribution-based competitive grouping’, and their basic function is to group players based on their contribution choices — i.e. high contributors with high contributors, and low contributors with low contributors. Theory predicts the following efficiency-equality tradeoff as a function of the mechanism’s inherent fuzziness: high levels of fuzziness should lead to maximal inefficiency, but perfect equality; decreasing fuzziness is predicted to improve efficiency, but at the cost of growing inequality. The main finding of our experimental investigation is that, contrary to tradeoff predictions, less fuzziness increases both efficiency and equality. In fact, these unambiguous welfare gains are partially realized already at levels where the mechanism is too fuzzy for any high-efficiency outcome to even be a Nash equilibrium.