Optimally Deceiving a Learning Leader in Stackelberg Games

Recent results have shown that algorithms for learning the optimal commitment in a Stackelberg game are susceptible to manipulation by the follower. These learning algorithms operate by querying the best responses of the follower, who consequently can deceive the algorithm by using fake best responses, typically by responding according to fake payoffs that are different from the actual ones. For this strategic behavior to be successful, the main challenge faced by the follower is to pinpoint the fake payoffs that would make the learning algorithm output a commitment that benefits them the most. While this problem has been considered before, the related literature has only focused on a simple setting where the follower can only choose from a finite set of payoff matrices, thus leaving the general version of the problem unanswered. In this paper, we fill this gap by showing that it is always possible for the follower to efficiently compute (near-)optimal fake payoffs, for various scenarios of learning interaction between the leader and the follower. Our results also establish an interesting connection between the follower’s deception and the leader’s maximin utility: through deception, the follower can induce almost any (fake) Stackelberg equilibrium if and only if the leader obtains at least their maximin utility in this equilibrium.

The Frequency of Convergent Games under Best-Response Dynamics

We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

The Carrot and the Stick

After I helped construct a school in Guatemala, I found their educational system might be corrupt. I used game theory, or the study of strategic interdependence, to analyze this issue. Each actor anticipates the other’s reactions and acts accordingly. Payoff matrices, 2-by-2 grids in which one actor’s options are listed across the top of the boxes and the second actor’s options are listed by the left side of the boxes, helped simplify this complex situation into a workable model that shows individual incentives and actions as well as the collective best action for societal welfare. If the first player makes “x” choice rather than “y” choice, is the second player incentivized to make “a” or “b” choice? Conversely, if the first player makes “y” choice, is the second player incentivized to make “a” or “b” choice? If, for a particular choice, no player can do better by unilaterally changing its decision, it is a Nash equilibrium. Ideally, the government won’t receive a bribe, and the parents will send their children to volunteer. The penalty if the local government gets caught accepting a bribe plays an important role; interestingly, increasing the government's goodwill amongst the local population through the school-building process would also lead to the ideal outcome. Further, simultaneously increasing both the penalty from accepting a bribe, as well as the goodwill to the government from school-building, produces the optimal outcome. I call this the “carrot and the stick” approach, and it may prove quite effective in ending corruption entirely.

Acceptable-and-attractive Approximate Solution of a Continuous Non-Cooperative Game on a Product of Sinusoidal Strategy Functional Spaces

A problem of solving a continuous noncooperative game is considered, where the player’s pure strategies are sinusoidal functions of time. In order to reduce issues of practical computability, certainty, and realizability, a method of solving the game approximately is presented. The method is based on mapping the product of the functional spaces into a hyperparallelepiped of the players’ phase lags. The hyperparallelepiped is then substituted with a hypercubic grid due to a uniform sampling. Thus, the initial game is mapped into a finite one, in which the players’ payoff matrices are hypercubic. The approximation is an iterative procedure. The number of intervals along the player’s phase lag is gradually increased, and the respective finite games are solved until an acceptable solution of the finite game becomes sufficiently close to the same-type solutions at the preceding iterations. The sufficient closeness implies that the player’s strategies at the succeeding iterations should be not farther from each other than at the preceding iterations. In a more feasible form, it implies that the respective distance polylines are required to be decreasing on average once they are smoothed with respective polynomials of degree 2, where the parabolas must be having positive coefficients at the squared variable.

Cooperative decision-making on fiscal and monetary policy in Iraq using the prisoner’s dilemma

This paper investigates the interaction between fiscal and monetary policy in Iraq after 2003 using the prisoner’s dilemma.The paper aims to determine the best form of coordination between these policies to achieve their goals; payoff matrix for both policies was constructed. To achieve the purpose, the quantitative approach was applied using several methods, including regression, building payoff matrices and decision analysis using a number of software.The results of the monetary policy payment function show that inflation rate has an inverse relationship with the auctions of selling foreign currency and a positive relationship with the government’s activity, while the fiscal policy function shows that real growth is positively related to price levels (the inverted Phillips curve) and correlates with the government’s activity. After using the Gambit Solution to determine the Nash balance, which is achieved through the expansion strategies of both policies to confirm the results, the Promethee-Gaia method was used for multi-criteria decision making. When the two policies interact with similar forces (50% each), the best decision is one of the expansionary strategies that help achieve their main objectives in the short and long term, represented by price stability and economic growth.The main conclusion is that the best way to achieve the goals of economic policy in Iraq is that the coordination of procedures between the two policies should be expansionary, since the Iraqi economy needs to be stimulated due to the under-exploitation of many its sectors, such as agriculture and industry.

Human-robot cooperation in economic games: People show strong reciprocity but conditional prosociality toward robots

Understanding the social dynamics in human-robot cooperation stands to broaden our understanding of human behaviour in general, as well as to improve the design of robots designated for certain social tasks. This study investigated human cooperative behaviours in the context of prisoner’s dilemma (PD) games and the extent to which people’s willingness to cooperate with a robot varies according to incentives provided by the game context. We manipulated the payoff matrices of human-robot PD games and predicted that people would cooperate more often in the situation where cooperating with the robot was a relatively more rewarding option. Although the pre-registered mixed effect logistic regression model showed no significant effect of incentive structure on human decisions in the iterated PD games overall, further exploratory analyses revealed that in early game rounds, participants made significantly more cooperative decisions when the game structure provided more incentives for cooperation. However, subsequent game decisions were dominantly driven by other two factors. The first was the robot’s previous game choices, where participants played a tit-for-tat strategy against the robot even though its decisions were random. Second, presentation of real-time game scores significantly impacted people’s cooperative tendencies. Participants only showed prosocial tendencies toward the robot when they had achieved high scores themselves. Our findings on the initial effects of incentive structure, robust reciprocity, and conditional prosociality in human-robot cooperation highlight ways in which we might expect social behaviour toward robots to differ from social behaviour toward humans, and help to establish the foundations necessary to support successful social collaboration with robots.

Triangular distribution and PERT method vs. payoff matrix for decision-making support in risk analysis of construction bidding: A case study

Decision-making in construction bidding represents a complex process due to the present risk. Risk or uncertainty cannot be ignored and should be treated as a constituent of decision-making. The paper aims to emphasize the importance of probability theory by comparing insufficiently applicable methods in practical bidding. The triangular distribution and the PERT method belong to three-point estimate techniques, while payoff matrices represent a multi-criteria approach. Also, selected methods belong to quantitative techniques for risk cost analysis. Still, the risk costs determination of the unit costs and the total costs of bids is often based on an intuitive approach. Therefore, compared results of the triangular distribution, PERT method, and payoff matrix techniques (minimin, minimax, expected monetary value, and expected opportunity loss) indicate the significance of risk costs estimating in tendering. The analysis of the results showed some overlaps in risk costs values obtained by the PERT method and expected monetary value technique. Those are due to the specificity of the chosen practical example and cannot be adopted as a rule. This means that selected methods and techniques are very useful for all bid estimation. The paper proved the complexity of decision-making, where the primary goal is to award a contract.

On Equilibrium Properties of the Replicator–Mutator Equation in Deterministic and Random Games

In this paper, we study the number of equilibria of the replicator–mutator dynamics for both deterministic and random multi-player two-strategy evolutionary games. For deterministic games, using Descartes’ rule of signs, we provide a formula to compute the number of equilibria in multi-player games via the number of change of signs in the coefficients of a polynomial. For two-player social dilemmas (namely the Prisoner’s Dilemma, Snow Drift, Stag Hunt and Harmony), we characterize (stable) equilibrium points and analytically calculate the probability of having a certain number of equilibria when the payoff entries are uniformly distributed. For multi-player random games whose pay-offs are independently distributed according to a normal distribution, by employing techniques from random polynomial theory, we compute the expected or average number of internal equilibria. In addition, we perform extensive simulations by sampling and averaging over a large number of possible payoff matrices to compare with and illustrate analytical results. Numerical simulations also suggest several interesting behaviours of the average number of equilibria when the number of players is sufficiently large or when the mutation is sufficiently small. In general, we observe that introducing mutation results in a larger average number of internal equilibria than when mutation is absent, implying that mutation leads to larger behavioural diversity in dynamical systems. Interestingly, this number is largest when mutation is rare rather than when it is frequent.