Fair Games Research Articles

The impact of randomness on the distribution of wealth: Some economic aspects of the Wright–Fisher diffusion process

In this paper we consider some elementary and fair zero-sum games of chance in order to study the impact of random effects on the wealth distribution of N interacting players. Even if an exhaustive analytical study of such games between many players may be tricky, numerical experiments highlight interesting asymptotic properties. In particular, we emphasize that randomness plays a key role in concentrating wealth in the extreme, in the hands of a single player. From a mathematical perspective, we interestingly adopt some diffusion limits for small and high-frequency transactions which are otherwise extensively used in population genetics. Finally, the impact of small tax rates on the preceding dynamics is discussed for several regulation mechanisms. We show that taxation of income is not sufficient to overcome this extreme concentration process in contrast to the uniform taxation of capital which stabilizes the economy and prevents agents from being ruined.

Quantitative fair simulation games

Simulation is an attractive alternative to language inclusion for automata as it is an under-approximation of language inclusion, but usually has much lower complexity. Simulation has also been extended in two orthogonal directions, namely, (1) fair simulation, for simulation over specified set of infinite runs; and (2) quantitative simulation, for simulation between weighted automata. While fair trace inclusion is PSPACE-complete, fair simulation can be computed in polynomial time. For weighted automata, the (quantitative) language inclusion problem is undecidable in general, whereas the (quantitative) simulation reduces to quantitative games, which admit pseudo-polynomial time algorithms.In this work, we study (quantitative) simulation for weighted automata with Büchi acceptance conditions, i.e., we generalize fair simulation from non-weighted automata to weighted automata. We show that imposing Büchi acceptance conditions on weighted automata changes many fundamental properties of the simulation games, yet they still admit pseudo-polynomial time algorithms.

Order statistics and the length of the best-n-of-(2n − 1) competition

ABSTRACTThe best-4-of-7 series is a popular playoff format to decide the champion in most North American professional sports. World Series (best-4-of-7) type competitions give rise to interesting probabilistic and statistical questions. We determine the expected length of this type of series by relating it to a problem involving order statistics. We also calculate the variance of the length and provide a simple formula for series of fair games. The method can be extended to derive higher order moments. This novel approach leads to new results that can be formulated in closed forms in terms of the distribution function of various binomial distributions. The emphasis is on establishing the connection to order statistics and obtaining closed forms. The relation to the negative binomial distribution as well as to the sooner waiting time problem in sequential testing is also discussed. We also consider the case when ties are allowed in the single games.

Winning fast in fair biased Maker-Breaker games

We study the (a:a) Maker-Breaker games played on the edge set of the complete graph on n vertices. In the following four games – perfect matching game, Hamiltonicity game, star factor game and path factor game, our goal is to determine the least number of moves which Maker needs in order to win these games. Moreover, for all games except for the star factor game, we show how Red can win in the strong version of these games.

The Uniqueness of Rock-Paper-Scissors-Lizard-Spock

SummaryWe show that the extension of Rock-Paper-Scissors to include Lizard and Spock is the unique such five move fair game up to isomorphism. We also categorize all analogous six move fair games.

How egalitarian are Nash equilibria in network cost-sharing games?

We consider the egalitarian social cost, which is the maximum individual cost (instead of the sum), when analyzing Nash equilibria in fair network cost-sharing games. Intuitively, the egalitarian price of anarchy reflects how uneven cost is distributed among players at equilibrium. We first show a tight upper bound of k in general fair network cost-sharing games, where k is the total number of players. For fair network cost-sharing games with a single source–sink pair and a relaxed benchmark, we then show an upper bound of n−1 on the egalitarian price of anarchy defined using such benchmark, where n is the network size. This gives a possibly better bound that does not depend on the number of players nor the costs.

The price of stability of fair undirected broadcast games is constant

Bounding the price of stability of undirected network design games with fair cost allocation is a challenging open problem in the Algorithmic Game Theory research agenda. Even though the generalization of such games in directed networks is well understood in terms of the price of stability (it is exactly H n , the n -th harmonic number, for games with n players), far less is known for network design games in undirected networks. The upper bound carries over to this case as well, while the best known lower bound is 2:245. For more restricted but interesting variants of such games, such as broadcast and multicast games, sublogarithmic upper bounds are known, while the best known lower bounds are 1:818 and 1:862, respectively. In this letter, we discuss a recent breakthrough in this field of research: an O (1) upper bound on the price of stability for undirected broadcast games.

Altruism and Its Impact on the Price of Anarchy

We study the inefficiency of equilibria for congestion games when players are (partially) altruistic. We model altruistic behavior by assuming that player i 's perceived cost is a convex combination of α i times his direct cost and α i times the social cost. Tuning the parameters α i allows smooth interpolation between purely selfish and purely altruistic behavior. Within this framework, we study primarily altruistic extensions of (atomic and nonatomic) congestion games, but also obtain some results on fair cost-sharing games and valid utility games. We derive (tight) bounds on the price of anarchy of these games for several solution concepts. Thereto, we suitably adapt the smoothness notion introduced by Roughgarden and show that it captures the essential properties to determine the robust price of anarchy of these games. Our bounds show that for atomic congestion games and cost-sharing games, the robust price of anarchy gets worse with increasing altruism, while for valid utility games, it remains constant and is not affected by altruism. However, the increase in the price of anarchy is not a universal phenomenon: For general nonatomic congestion games with uniform altruism, the price of anarchy improves with increasing altruism. For atomic and nonatomic symmetric singleton congestion games, we derive bounds on the pure price of anarchy that improve as the average level of altruism increases. (For atomic games, we only derive such bounds when cost functions are linear.) Since the bounds are also strictly lower than the robust price of anarchy, these games exhibit natural examples in which pure Nash equilibria are more efficient than more permissive notions of equilibrium.

The athlete biological passport: ticket to a fair Commonwealth Games.

In summer 2014, the world watched as Glasgow hosted the 2014 Commonwealth Games and athletes pushed the boundaries of human performance. Sport has developed into a multi-billion pound industry leading to the development of a 'win at any cost' mentality in some individuals. The abuse of performance-enhancing drugs has developed into a sophisticated arms race between those unfairly enhancing performance and those wishing to preserve the dignity of sport and the health of the competitors. The challenge for the Commonwealth games organising committee was to ensure that competition remained fair and that athletes were kept safe. The athlete biological passport is a system implemented by the World Anti-Doping Agency directed towards enhancing the identification of those athletes accountable for the misuse of performance-enhancing substances. This article exemplifies which drugs are currently being exploited and how the athlete biological passport has evolved to improve their detection.

The Price of Uncertainty

In this work, we study the degree to which small fluctuations in costs in well-studied potential games can impact the result of natural best-response and improved-response dynamics. We call this the Price of Uncertainty and study it in a wide variety of potential games including fair cost-sharing games, set-cover games, routing games, and job-scheduling games. We show that in certain cases, even extremely small fluctuations can have the ability to cause these dynamics to spin out of control and move to states of much higher social cost, whereas in other cases these dynamics are much more stable even to large degrees of fluctuation. We also consider the resilience of these dynamics to a small number of Byzantine players about which no assumptions are made. We show again a contrast between different games. In certain cases (e.g., fair cost-sharing, set-cover, job-scheduling) even a single Byzantine player can cause best-response dynamics to transition from low-cost states to states of substantially higher cost, whereas in others (e.g., the class of β -nice games, which includes routing, market-sharing and many others) these dynamics are much more resilient. Overall, our work can be viewed as analyzing the inherent resilience or safety of games to different kinds of imperfections in player behavior, player information, or in modeling assumptions made.

Gambler’s ruin and winning a series by m games

Two teams play a series of games until one team accumulates m more wins than the other. These series are fairly common in some sports provided that the competition has already extended beyond some number of games. We generalize these schemes to allow ties in the single games. Different approaches offer different advantages in calculating the winning probabilities and the distribution of the duration N, including difference equations, conditioning, explicit and implicit path counting, generating functions and a martingale-based derivation of the probability and moment generating functions of N. The main result of the paper is the determination of the exact distribution of N for a series of fair games without ties as a sum of independent geometrically distributed random variables and its approximation.

Probably not: future prediction using probability and statistical inference

Introduction. 1. An Introduction to Probability. Predicting The Future. Rule Making. Random Events and Probability. The Lottery {Very Improbable Events and Large Data Sets}. Coin Flipping {Fair Games, Looking Backwards For Insight}. The Coin Flip Strategy That Can't Lose. The Prize Behind The Door {Looking Backwards For Insight, Again}. The Checker Board {Dealing With Only Part Of The Data Set}. 2. Probability Distribution Functions And Some Basics. The Probability Distribution Function. Averages And Weighted Averages. Expected Values. The Basic Coin Flip Game. The Standard Deviation. The Cumulative Distribution Function. The Confidence Interval. Final Points. 3. Building a Bell. 4. Random Walks. The One Dimensional Random Walk. What Probability Really Means. Diffusion. 5. Life Insurance and Social Security. Insurance as Gambling. Life Tables. Birth Rates and Population Stability. Life Tables, Again. Premiums. Social Security - Sooner Or Later?. 6. Binomial Probabilities. The Binomial Probability Formula. Permutations And Combinations. Large Number Approximations. The Poisson Distribution. Disease Clusters. Clusters. 7. Pseudorandom Numbers and Monte Carlo Simulations. Pseudorandom Numbers. The Middle Square PSNG. The Linear Congruential PSNG. A Normal Distribution Generator. An Arbitrary Distribution Generator. Monte Carlo Simulations. A League Of Our Own. 8. Some Gambling Games In Detail. The Basic Coin Flip Game. The Gantt Chart. The Ultimate Winning Strategy. The Game Show. Parimutuel Betting. 9. Traffic Lights And Traffic. Outsmarting A Traffic Light?. Many Lights And Many Cars. Simulating Traffic Flow - The Simulation. Simulation Results. 10. Combined And Conditional Probabilities. Functional Notation. Conditional Probability. Medical Test Results. The Shared Birthday Problem. 11. Scheduling And Waiting. Scheduling Appointments In The Doctor's Office. Lunch With A Friend. Waiting For A Bus. 12. Stock Market Portfolios. 13. Benford, Parrondo and Simpson. Benford's Law. Parrondo's Paradox. Simpson's Paradox. 14. Networks, Infectious Disease Propagation and Chain Letters. Degrees Of Separation. Propagation Along Networks. Some Other Uses Of Networks. Neighborhood Chains. 15. Bird Counting. A Walk In The Woods. A Model Of Bird Flying Habits. Spotting A Bird. Putting It All Together. 16. Statistical Mechanics And Heat. Statistical Mechanics. Thermodynamics. 17. Introduction To Statistical Analysis. Sampling. Sample Distributions and Standard Deviations. Estimating Population Average From A Sample. The Student-T Distribution. Polling Statistics. Did A Sample Come From A Given Population?. 18. Chaos and Quanta. Chaos. Probability In Quantum Mechanics.

Illusion of control in a Brownian game

Both single-player Parrondo games (SPPG) and multi-player Parrondo games (MPPG) display the Parrondo effect (PE) wherein two or more individually fair (or losing) games yield a net winning outcome if alternated periodically or randomly. (There is a more formal, less restrictive definition of the PE.) We illustrate that, when subject to an elementary optimization rule, the PG displays degraded rather than enhanced returns. Optimization provides only the illusion of control, when low-entropy strategies (i.e., which use more information) under-perform random strategies (with maximal entropy). This illusion is unfortunately widespread in many human attempts to manage or predict complex systems. For the PG, the illusion is especially striking in that the optimization rule reverses an already paradoxical-seeming positive gain—the Parrondo effect proper—and turns it negative. While this phenomenon has been previously demonstrated using somewhat artificial conditions in the MPPG [L. Dinis, J.M.R. Parrondo, Europhys. Lett. 63 (2003) 319; J.M.R. Parrondo, L. Dinis, J. Buceta, K. Lindenberg, Advances in Condensed Matter and Statistical Mechanics, E. Korutcheva, R. Cuerno (Eds.), Nova Science Publishers, New York, 2003], we demonstrate it in the natural setting of a history-dependent SPPG.

Reversals of chance in paradoxical games

We present two collective games with new paradoxical features when they are combined. Besides reproducing the so-called Parrondo effect, where a winning game is obtained from the alternation of two fair games, there also exists a current inversion when varying the mixing probability between the games. We show that this is a new effect insofar one of the games is an unbiased random walk without internal structure. We present a detailed study by means of a discrete-time Markov chain analysis, obtaining analytical expressions for the stationary probabilities for a finite number of players. We also provide qualitative insight into this current inversion effect.

On Parrondo's paradox: how to construct unfair games by composing fair games

AbstractWe construct games of chance from simpler games of chance. We show that it may happen that the simpler games of chance are fair or unfavourable to a player and yet the new combined game is favourable—this is a counter-intuitive phenomenon known as Parrondo's paradox. We observe that all of the games in question are random walks in periodic environments (RWPE) when viewed on the proper time scale. Consequently, we use RWPE techniques to derive conditions under which Parrondo's paradox occurs.

ON ASTUMIAN'S PARADOX

An Astumian game is defined by a finite Markov chain with state space S with precisely two absorbing states, the winning and the losing state. The other states are transient, one of them is the starting position. The game is said to be losing (respectively fair respectively winning) if the probability to be absorbed at the winning state is smaller than 0.5 (respectively = 0.5 respectively larger than 0.5). Astumian's paradox states that there are losing games on the same state space S a stochastic mixture of which is winning. (By "stochastic mixture" we mean that in each step one decides with the help of a fair coin whether to use the transition probabilities of the first or the second game.) Most of our results concern fair games. Mixtures are systematically investigated. Rather surprisingly, the winning probability of the mixture of fair games can be arbitrarily close to zero (or to one). Even more counter-intuitive are examples of definitely losing games (this means that the winning probability is exactly zero) such that the winning probability of the mixture is arbitrarily close to one. We show, however, that such extreme examples are possible only if one tolerates huge running times of the game. As a natural generalization one can also consider arbitrary mixtures: the fair coin is replaced by a biased one, with probability λ respectively 1-λ one plays with the first respectively the second game. It turns out that fair games exist such that — depending on the choice of λ — the λ-mixture can be fair, losing or winning.

Rethinking Fair Games

Take a Moment to Think About how you would answer this question: What does it mean for a game to be fair? When asked this question, many people focus on playing fair or the fact that neither player cheats.

Parrondo's games as a discrete ratchet

We write the master equation describing the Parrondo's games as a consistent discretization of the Fokker–Planck equation for an overdamped Brownian particle describing a ratchet. Our expressions, besides giving further insight on the relation between ratchets and Parrondo's games, allow us to precisely relate the games probabilities and the ratchet potential such that periodic potentials correspond to fair games and winning games produce a tilted potential.

Learning to bid – an experimental study of bid function adjustments in auctions and fair division games

We examine learning behaviour in auction and fair division experiments with independent private values under two different price rules, first and second price. Participants play all four games repeatedly and submit complete bid functions rather than single bids. This allows us to study how institutional changes are anticipated and whether learning is influenced by the structural differences between games. We find that learning does not drive bidding towards the benchmark solution. Bid functions are adjusted globally rather than locally. Directional learning theory offers a partial explanation for bid changes. The data support a cognitive approach to learning.

Equilibrium valuation of illiquid assets

We develop an equilibrium model of illiquid asset valuation based on search and matching. We propose several measures of illiquidity and show how these measures behave. We also show that the equilibrium amount of search may be less than, equal to or greater than the amount of search that is socially optimal. Finally, we show that excess returns on illiquid assets are fair games if returns are defined to include the appropriate shadow prices.