Linear Payoff Research Articles

Unbending strategies shepherd cooperation and suppress extortion in spatial populations

Evolutionary game dynamics on networks typically consider the competition among simple strategies such as cooperation and defection in the Prisoner’s Dilemma and summarize the effect of population structure as network reciprocity. However, it remains largely unknown regarding the evolutionary dynamics involving multiple powerful strategies typically considered in repeated games, such as the zero-determinant (ZD) strategies that are able to enforce a linear payoff relationship between them and their co-players. Here, we consider the evolutionary dynamics of always cooperate (AllC), extortionate ZD (extortioners), and unbending players in lattice populations based on the commonly used death-birth updating. Out of the class of unbending strategies that can foster reciprocal cooperation and fairness among extortionate players, we consider a particular candidate, pre-optimized through the machine-learning method of particle swarm optimization (PSO), called PSO Gambler. We derive analytical results under weak selection and rare mutations, including pairwise fixation probabilities and long-term frequencies of strategies. In the absence of the third unbending type, extortioners can achieve a half-half split in equilibrium with unconditional cooperators for sufficiently large extortion factors. However, the presence of unbending players fundamentally changes the dynamics and tilts the system to favor unbending cooperation. Most surprisingly, extortioners cannot dominate at all regardless of how large their extortion factor is, and the long-term frequency of unbending players is maintained almost as a constant. Our analytical method is applicable to studying the evolutionary dynamics of multiple strategies in structured populations. Our work provides insights into the interplay between network reciprocity and direct reciprocity, revealing the role of unbending strategies in enforcing fairness and suppressing extortion.

Relatedness in zero-determinant strategies

Relatedness is necessary and causal in the development of social life. Interlayer relatedness is a measure of how one player’s decisions affect the decisions of other players in the game. The relatedness can be positive or negative. We had to determine how effective each strategy was under specific conditions, and how the correlation between players affected their payoffs. In this paper, we analytically study the strategies that enforce linear payoff relationships in the Iterated Prisoner’s Dilemma (IPD) game considering both a relatedness factor. As a result, we first reveal that the payoffs of two players and three players can be represented by the form of determinants as shown by Press and Dyson even with the factor.

Reputation-based synergy and discounting mechanism promotes cooperation

A good group reputation often facilitates more efficient synergistic teamwork in production activities. Here we translate this simple motivation into a reputation-based synergy and discounting mechanism in the public goods game. Specifically, the reputation type of a group, either good or bad determined by a reputation threshold, modifies the nonlinear payoff structure described by a unified reputation impact factor. Results show that this reputation-based incentive mechanism could effectively promote cooperation compared with linear payoffs, despite the coexistence of synergy and discounting effects. Notably, the complicated interactions between reputation impact and reputation threshold result in a sharp phase transition from full cooperation to full defection. We also find that the presence of a few discounting groups could increase the average payoffs of cooperators, leading to an interesting phenomenon that when the reputation threshold is raised, the gap between the average payoffs of cooperators and defectors increases while the overall payoff decreases. We further extend our framework to heterogeneous situations and show how the variability of individuals affect the evolutionary outcomes. Our work provides important insights into facilitating cooperation in social groups.

A unified formula of the optimal portfolio for piecewise hyperbolic absolute risk aversion utilities

We propose a general family of piecewise hyperbolic absolute risk aversion (PHARA) utilities, including many classic and non-standard utilities as examples. A typical application is the composition of a HARA preference and a piecewise linear payoff in asset allocation. We derive a unified closed-form formula of the optimal portfolio, which is a four-term division. The formula has clear economic meanings, reflecting the behavior of risk aversion, risk seeking, loss aversion and first-order risk aversion. We conduct a general asymptotic analysis to the optimal portfolio, which directly serves as an analytical tool for financial analysis. We compare this PHARA portfolio with those of other utility families both analytically and numerically. One main finding is that risk-taking behaviors are greatly increased by non-concavity and reduced by non-differentiability of the PHARA utility. Finally, we use financial data to test the performance of the PHARA portfolio in the market.

Approximating the Value of Zero-Sum Differential Games with Linear Payoffs and Dynamics

We consider two-player zero-sum differential games of fixed duration, where the running payoff and the dynamics are both linear in the controls of the players. Such games have a value, which is determined by the unique viscosity solution of a Hamilton–Jacobi-type partial differential equation. Approximation schemes for computing the viscosity solution of Hamilton–Jacobi-type partial differential equations have been proposed that are valid in a more general setting, and such schemes can of course be applied to the problem at hand. However, such approximation schemes have a heavy computational burden. We introduce a discretized and probabilistic version of the differential game, which is straightforward to solve by backward induction, and prove that the solution of the discrete game converges to the viscosity solution of the partial differential equation, as the discretization becomes finer. The method removes part of the computational burden of existing approximation schemes.

The age-performance relationship for a cognitive-intensive task: Empirical evidence from chess grandmasters

To gain insight into the age-performance relationship (APR) for cognitively demanding tasks, we analyse the APR for the task of playing chess from a theoretical and empirical perspective. We set up a game-theoretic model showing that for risk-neutral players who are homogeneous in their linear payoffs, the APR can be estimated with either players' game outcomes or their Elo ratings. This result is empirically substantiated with data on outcomes of games played at an annual international chess tournament (1970–2021), and with players' Elo ratings. Further, the findings support the estimation of the APR with Elo ratings using a model that controls for player fixed effects and period random effects, which is also in accordance with the theoretical model. Next, we show that on average chess performance peaks in grandmasters’ early-thirties and declines thereafter. Implications of the findings for a labour market characterised by an increase in cognitively demanding jobs and individuals working longer are discussed.

Risk-averse Contextual Multi-armed Bandit Problem with Linear Payoffs

In this paper we consider the contextual multi-armed bandit problem for linear payoffs under a risk-averse criterion. At each round, contexts are revealed for each arm, and the decision maker chooses one arm to pull and receives the corresponding reward. In particular, we consider mean-variance as the risk criterion, and the best arm is the one with the largest mean-variance reward. We apply the Thompson sampling algorithm for the disjoint model, and provide a comprehensive regret analysis for a variant of the proposed algorithm. For T rounds, K actions, and d-dimensional feature vectors, we prove a regret bound of \(O\left({\left({1 + \rho + {1 \over \rho}} \right)d\,\ln \,T\ln {K \over \delta}\sqrt {dK{T^{1 + 2}}\ln {K \over \delta}{1 \over}}} \right)\) that holds with probability 1 − δ under the mean-variance criterion with risk tolerance ρ, for any \(0 < \in < \frac{1}{2},0 < \delta < 1\). The empirical performance of our proposed algorithms is demonstrated via a portfolio selection problem.

Intensive care unit/step-down unit queuing game with length of stay decisions

In this paper, length of stay (LOS) competition between two servers in tandem without buffer between them is investigated using queuing games. This system typifies the relationship between the intensive care unit (ICU) and the step-down unit (SDU) of a hospital. We model and analyze the equilibrium LOS decision under four different games (one cooperative and three non-cooperative games) as follows: (i) both servers cooperate; (ii) the servers do not cooperate and make decisions simultaneously; (iii) the servers do not cooperate and the first server, the ICU, is the leader (ICU Stackelberg); (iv) the servers do not cooperate and the second server, the SDU, is the leader (SDU Stackelberg). The payoff of the ICU is expressed as the difference between the service benefit and the waiting in queue penalty, while that of the SDU is the difference between the service benefit and the overstay penalty. The results show that LOS decisions of each server depends critically on the payoff function’s form and the exogenous demand. Secondly, with a linear payoff function, the SDU is only beneficial to the system if the unit cost is greater than the unit reward at the ICU. Our results revealed also that payoffs depend on the substitutability in both ICU Stackelberg and SDU Stackelberg games. When most of the LOS is spent at the ICU unit. Our results suggest that the critical care pathway performs better under coordination and or leadership at the ICU level.

Adapting paths against zero-determinant strategies in repeated prisoner’s dilemma games

Long-term cooperation, competition, or exploitation among individuals can be modeled through repeated games. In repeated games, Press and Dyson discovered zero-determinant (ZD) strategies that enforce a special relationship between two players. This special relationship implies that a ZD player can unilaterally impose a linear payoff relationship to the opponent regardless of the opponent’s strategies. A ZD player also has a property that can lead the opponent to an unconditional cooperation if the opponent tries to improve its payoff. This property has been mathematically confirmed by Chen and Zinger. Humans often underestimate a payoff obtained in the future. However, such discounting was not considered in their analysis. Here, we mathematically explored whether a ZD player can lead the opponent to an unconditional cooperation even if a discount factor is incorporated. Specifically, we represented the expected payoff with a discount factor as the form of determinants and calculated whether the values obtained by partially differentiating each factor in the strategy vector become positive. As a result, we proved that the strategy vector ends up as an unconditional cooperation even when starting from any initial strategy. This result was confirmed through numerical calculations. We extended the applicability of ZD strategies to real world problems.

Moneyness, Underlying Asset Volatility, and the Cross-Section of Option Returns

Abstract We study the effect of an asset’s volatility on the expected returns of European options on the asset. Deriving predictions from a stochastic discount factor model, we show that the effect depends on whether variations in the asset’s volatility are driven by systematic or idiosyncratic volatility. While idiosyncratic-volatility-induced variations only affect the option elasticity, systematic-volatility-induced variations also oppositely affect the expected return of the asset. Since the expected asset return (elasticity) effect dominates for options with more linear (non-linear) payoffs, systematic volatility prices sufficiently in-the-money (out-of-the-money) options with the opposite (same) sign as idiosyncratic volatility. Using single-stock calls as test assets, double-sorted portfolios and Fama–MacBeth (1973) regressions broadly support the model’s predictions.

Dynkin Games with Incomplete and Asymmetric Information

We study the value and the optimal strategies for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our Bayesian setup, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result that allows us to find the value of the game and players’ optimal strategies by solving suitable quasi-variational inequalities with some nonstandard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained.

Conditions for the existence of zero-determinant strategies under observation errors in repeated games

Repeated games are useful models to analyze long term interactions of living species and complex social phenomena. Zero-determinant (ZD) strategies in repeated games discovered by Press and Dyson in 2012 enforce a linear payoff relationship between a focal player and the opponent. This linear relationship can be set arbitrarily by a ZD player. Hence, a subclass of ZD strategies can fix the opponent’s expected payoff and another subclass of the strategies can exceed the opponent for the expected payoff. Since this discovery, theories for ZD strategies are extended to cope with various natural situations. It is especially important to consider the theory of ZD strategies for repeated games with a discount factor and observation errors because it allows the theory to be applicable in the real world. Recent studies revealed their existence of ZD strategies even in repeated games with both factors. However, the conditions for the existence has not been sufficiently analyzed. Here, we mathematically analyzed the conditions in repeated games with both factors. First, we derived the thresholds of a discount factor and observation errors which ensure the existence of Equalizer and positively correlated ZD (pcZD) strategies, which are well-known subclasses of ZD strategies. We found that ZD strategies exist only when a discount factor remains high as the error rates increase. Next, we derived the conditions for the expected payoff of the opponent enforced by Equalizer as well as the conditions for the slope and base line payoff of linear lines enforced by pcZD. As a result, we found that, as error rates increase or a discount factor decreases, the conditions for the linear line that Equalizer or pcZD can enforce become strict.

Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems

In this paper we address game theory problems arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problems addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of a Nash game. The resulting problems are nonconvex QP ones with linear constraints and turn out to be very challenging. We will show that the most recent approaches for the minimization of nonconvex QP functions over polytopes, including commercial solvers such as CPLEX and GUROBI, are unable to solve to optimality even test instances with n=50 variables. For this reason, we propose to extend with them the current benchmark set of test instances for QP problems. We also present a spatial branch-and-bound approach for the solution of these problems, where a predominant role is played by an optimality-based domain reduction, with multiple solutions of LP problems at each node of the branch-and-bound tree. Of course, domain reductions are standard tools in spatial branch-and-bound approaches. However, our contribution lies in the observation that, from the computational point of view, a rather aggressive application of these tools appears to be the best way to tackle the proposed instances. Indeed, according to our experiments, while they make the computational cost per node high, this is largely compensated by the rather slow growth of the number of nodes in the branch-and-bound tree, so that the proposed approach strongly outperforms the existing solvers for QP problems.

Near-Optimal Regret Bounds for Contextual Combinatorial Semi-Bandits with Linear Payoff Functions

The contextual combinatorial semi-bandit problem with linear payoff functions is a decision-making problem in which a learner chooses a set of arms with the feature vectors in each round under given constraints so as to maximize the sum of rewards of arms. Several existing algorithms have regret bounds that are optimal with respect to the number of rounds T. However, there is a gap of Õ(max(√d, √k)) between the current best upper and lower bounds, where d is the dimension of the feature vectors, k is the number of the chosen arms in a round, and Õ(·) ignores the logarithmic factors. The dependence of k and d is of practical importance because k may be larger than T in real-world applications such as recommender systems. In this paper, we fill the gap by improving the upper and lower bounds. More precisely, we show that the C2UCB algorithm proposed by Qin, Chen, and Zhu (2014) has the optimal regret bound Õ(d√kT + dk) for the partition matroid constraints. For general constraints, we propose an algorithm that modifies the reward estimates of arms in the C2UCB algorithm and demonstrate that it enjoys the optimal regret bound for a more general problem that can take into account other objectives simultaneously. We also show that our technique would be applicable to related problems. Numerical experiments support our theoretical results and considerations.

Zero-Determinant Strategies in Repeated Multiplayer Social Dilemmas With Discounted Payoffs

In two-player repeated games, zero-determinant (ZD) strategies enable a player to unilaterally enforce a linear payoff relation between her own and her opponent's payoff irrespective of the opponent's strategy. This manipulative nature of the ZD strategies attracted significant attention from researchers due to their close connection to controlling distributively the outcome of evolutionary games in large populations. In this article, necessary and sufficient conditions are derived for a payoff relation to be enforceable in multiplayer social dilemmas with a finite expected number of rounds that is determined by a fixed and common discount factor. Thresholds exist for such a discount factor above which desired payoff relations can be enforced. Our results show that depending on the group size and the ZD strategist's initial probability to cooperate their existing extortionate, generous, and equalizer ZD strategies. The threshold discount factors rely on the desired payoff relation and the variation in the single-round payoffs. To show the utility of our results, we apply them to multiplayer social dilemmas, and show how discounting affects ZD Nash equilibria.

Effects of synergy and discounting on cooperation in spatial public goods games

Hauert et al. [1] first observed the effects of discounting and synergy in social dilemma games and systematically studied these effects on cooperation in an unstructured population. This paper extends the effects of discounting and synergy into spatial public goods games (PGG) with a population structure. Specifically, an economy-scale threshold and an amplitude parameter to describe conditions and degrees of discounting and synergy are introduced, respectively. Simulation results show that nonlinear payoff based on synergy and discounting effects can promote cooperation more significantly than linear payoff structure. Moreover, the smaller the threshold value, the easier to trigger the synergy effect and remove the discounting effect; the larger the amplitude parameter, the stronger the effect of synergy and discounting to play a role. In both situations, it is more favorable for the emergence and prosperity of cooperation. These results are helpful to understand the evolution of cooperation in nonlinear PGG.

Rational Models for Inflation-Linked Derivatives

We construct models for the pricing and risk management of inflation-linked derivatives. The models are rational in the sense that linear payoffs written on the consumer price index have prices that are rational functions of the state variables. The nominal pricing kernel is constructed in a multiplicative manner that allows for closed-form pricing of vanilla inflation products suchlike zero-coupon swaps, year-on-year swaps, caps and floors, and the exotic limited-price-index swap. We study the conditions necessary for the multiplicative nominal pricing kernel to give rise to short rate models for the nominal interest rate process. The proposed class of pricing kernel models retains the attractive features of a nominal multi-curve interest rate model, such as closed-form pricing of nominal swaptions, and it isolates the so-called inflation convexity-adjustment term arising from the covariance between the underlying stochastic drivers. We conclude with examples of how the model can be calibrated to EUR data.

Zero-determinant strategies under observation errors in repeated games.

Zero-determinant (ZD) strategies are a novel class of strategies in the repeated prisoner's dilemma (RPD) game discovered by Press and Dyson. This strategy set enforces a linear payoff relationship between a focal player and the opponent regardless of the opponent's strategy. In the RPD game, games with discounting and observation errors represent an important generalization, because they are better able to capture real life interactions which are often noisy. However, they have not been considered in the original discovery of ZD strategies. In some preceding studies, each of them has been considered independently. Here, we analytically study the strategies that enforce linear payoff relationships in the RPD game considering both a discount factor and observation errors. As a result, we first reveal that the payoffs of two players can be represented by the form of determinants as shown by Press and Dyson even with the two factors. Then, we search for all possible strategies that enforce linear payoff relationships and find that both ZD strategies and unconditional strategies are the only strategy sets to satisfy the condition. We also show that neither Extortion nor Generous strategies, which are subsets of ZD strategies, exist when there are errors. Finally, we numerically derive the threshold values above which the subsets of ZD strategies exist. These results contribute to a deep understanding of ZD strategies in society.

Subjective Beliefs in International Agreements

We study the impact of countries’ subjective beliefs, i.e., pessimism and optimism, on international agreements to provide global public goods. Under linear payoffs, we find that while pessimism could decrease signatories’ efforts, it can also increase the coalition size. Optimism yields the opposite effects. Using an objective welfare criterion, signatories are typically worse off but non-signatories could be better off. As a result, despite the bias the social welfare may increase. When some countries have identical subjective beliefs (optimistic or pessimistic) and the rest have objective beliefs, the proportion of subjective signatories strongly affects signatories’ effort level. A stable coalition may consist of countries with one type of beliefs or a mixture of types. If it consists of only type, then this type is the relatively more optimistic type.

Nonzero-Sum Stochastic Differential Games Between an Impulse Controller and a Stopper

We study a two-player nonzero-sum stochastic differential game, where one player controls the state variable via additive impulses, while the other player can stop the game at any time. The main goal of this work is to characterize Nash equilibria through a verification theorem, which identifies a new system of quasivariational inequalities, whose solution gives equilibrium payoffs with the correspondent strategies. Moreover, we apply the verification theorem to a game with a one-dimensional state variable, evolving as a scaled Brownian motion, and with linear payoff and costs for both players. Two types of Nash equilibrium are fully characterized, i.e. semi-explicit expressions for the equilibrium strategies and associated payoffs are provided. Both equilibria are of threshold type: in one equilibrium players’ intervention are not simultaneous, while in the other one the first player induces her competitor to stop the game. Finally, we provide some numerical results describing the qualitative properties of both types of equilibrium.