We examine the random allocation problem and, adhering to the principle of favoring higher rank, propose an alternative extension to the probabilistic setting, which does not rely on welfare criteria, to eliminate an unfair scenario where agents who rank objects higher may end up receiving more better objects ex-post also. We introduce the property of interim favoring support, which is satisfied by the adaptive Boston Mechanism. Additionally, we propose a new fairness criterion, termed equal support equal claim, which further characterizes the adaptive Boston Mechanism.

Furthermore, we incorporate player 0 into the random allocation framework. This player, who may act as a social planner, manager, or mediator, does not receive any objects but holds incomplete preferences over the resulting allocations based on the full ordinal preference profile including player 0. We present two guiding principles to clarify the conditions under which the social planner’s opinion cannot be dismissed and when an agent’s opinion must be respected. The first principle respecting social planner, Make-Full-Use-Efficiently (MFUE), asserts that for any given object, no agent with a stronger bias should receive the object before those with weaker bias are satisfied, which strengthens the notion of ordinal efficiency. The second principle respecting agents, Equal-Bias-Equal-Treatment (EBET), requires that agents in identical bias positions be treated equally. To fulfil these two fairness principles, we develop a method to precisely characterize them.

Finally, we introduce a new efficiency concept, interim efficiency, which is stronger than ex-post Pareto efficiency but weaker than ordinal efficiency. We construct the Random Flow mechanism to achieve interim efficiency. Experimental analysis shows that Random Flow results in less envy across preference profiles compared to the Random Priority mechanism.

Abstract:

This study explores the prevalence of strictly dominated strategies in random games, highlighting two primary contributions: computational and economic. We establish that as the dimensions of a game increase, in a manner that is not too lopsided, the likelihood of the absence of a strictly dominated strategy approaches unity. This is particularly ev-ident in large square games where the row size is proportionate to the column size within a specific range. Consequently, the probability of a game being dominance solvable di-minishes to zero. Moreover, we demonstrate the results are very nearly tight: deviations from it result in a non-zero probability of encountering a strictly dominated strategy. Our findings initially emphasize the significance of the parameters in the underlying probability distribution.

Additionally, we discover the behaviour of the fixed portion, q, of dominated strategies when M, N goes to infinity. Specifically, for row player, we show the probability will go to 0 as M, N goes to infinity with M ≤ N/q and the probability will go to 1 as M, N go to infinity with M ≥ N^N.

Lastly, we introduce an efficient algorithm that significantly reduces computational complexity. This algorithm, an improvement over the conventional approach, decreases the time complexity from O(M^2 N ) to O(M N ), where M and N represent the row and column sizes, respectively. This advancement is accompanied by a marginal upper bound error denoted as o(M, N ), enhancing the practicality of analyzing strictly dominated strategies in large-scale games

Abstract:

This paper aims to provide a representation theorem for consumption stream in the simplex in continuous time, by using the weaker axiom of ”Agree with”, the strong axiom of “Preferential Independence”, and the axioms of “Separability” and “Measurability” on the simplex, consistent with the spirit of Koopman’s work between 1960-1972, and Epstein’s work in 1983 and 2003. Moreover, it considers a model containing piecewise continuous trajectories on the simplex, such as a portfolio optimization problem, a cake-cutting problem, or a maximization problem with a fixed budget (i,e., time, asset, probability distributions, etc.), which also allows some discontinuous trajectories. 

Abstract:

Theoretically, this work relaxes the rationality behind IESDS in the normal-form game and Backward Induction in the extensive-form game, namely common p-believe in rationality in the normal-form game and common p-belief in the opponent’s future rationality in the extensive-form game. Exper- imentally, this work uses the experimental data from Masaki, Guillaume, and Sevgi (2023)[1] to verify the result.

An individual would likely be happier avoiding constant comparisons. To avoid perceived inequality from comparison, envy-freeness is widely used in economics.  However, individuals often compare both their inputs and outcomes with those of others and then respond to eliminate any perceived inequities. More precisely, even if agents receive the same output, they will feel perceived inequities when they have different inputs as preferences. We generalize this idea to compare situations where agents' inputs and outcomes differ. We then interpret the concept of Jealousy-Free and introduce this property to a random allocation problem. Specifically, an agent feels unfair if others receive a larger share of their highest-ranked object while they do not. However, disappointment arises when we introduce sd-strategy-proofness that truth-telling is a weak dominant strategy for every agent, as it then shrinks to equal division.