34 research outputs found
Fair Resource Sharing with Externailities
We study a fair resource sharing problem, where a set of resources are to be
shared among a set of agents. Each agent demands one resource and each resource
can serve a limited number of agents. An agent cares about what resource they
get as well as the externalities imposed by their mates, whom they share the
same resource with. Apparently, the strong notion of envy-freeness, where no
agent envies another for their resource or mates, cannot always be achieved and
we show that even to decide the existence of such a strongly envy-free
assignment is an intractable problem. Thus, a more interesting question is
whether (and in what situations) a relaxed notion of envy-freeness, the Pareto
envy-freeness, can be achieved: an agent i envies another agent j only when i
envies both the resource and the mates of j. In particular, we are interested
in a dorm assignment problem, where students are to be assigned to dorms with
the same capacity and they have dichotomous preference over their dorm-mates.
We show that when the capacity of the dorms is 2, a Pareto envy-free assignment
always exists and we present a polynomial-time algorithm to compute such an
assignment; nevertheless, the result fails to hold immediately when the
capacities increase to 3, in which case even Pareto envy-freeness cannot be
guaranteed. In addition to the existential results, we also investigate the
implications of envy-freeness on proportionality in our model and show that
envy-freeness in general implies approximations of proportionality
Envy-freeness in house allocation problems
We consider the house allocation problem, where m houses are to be assigned to n agents so that each agent gets exactly one house. We present a polynomial-time algorithm that determines whether an envy-free assignment exists, and if so, computes one such assignment. We also show that an envy-free assignment exists with high probability if the number of houses exceeds the number of agents by a logarithmic factor
On the Inducibility of Stackelberg Equilibrium for Security Games
Strong Stackelberg equilibrium (SSE) is the standard solution concept of
Stackelberg security games. As opposed to the weak Stackelberg equilibrium
(WSE), the SSE assumes that the follower breaks ties in favor of the leader and
this is widely acknowledged and justified by the assertion that the defender
can often induce the attacker to choose a preferred action by making an
infinitesimal adjustment to her strategy. Unfortunately, in security games with
resource assignment constraints, the assertion might not be valid; it is
possible that the defender cannot induce the desired outcome. As a result, many
results claimed in the literature may be overly optimistic. To remedy, we first
formally define the utility guarantee of a defender strategy and provide
examples to show that the utility of SSE can be higher than its utility
guarantee. Second, inspired by the analysis of leader's payoff by Von Stengel
and Zamir (2004), we provide the solution concept called the inducible
Stackelberg equilibrium (ISE), which owns the highest utility guarantee and
always exists. Third, we show the conditions when ISE coincides with SSE and
the fact that in general case, SSE can be extremely worse with respect to
utility guarantee. Moreover, introducing the ISE does not invalidate existing
algorithmic results as the problem of computing an ISE polynomially reduces to
that of computing an SSE. We also provide an algorithmic implementation for
computing ISE, with which our experiments unveil the empirical advantage of the
ISE over the SSE.Comment: The Thirty-Third AAAI Conference on Artificial Intelligenc
Sequential Principal-Agent Problems with Communication: Efficient Computation and Learning
We study a sequential decision making problem between a principal and an
agent with incomplete information on both sides. In this model, the principal
and the agent interact in a stochastic environment, and each is privy to
observations about the state not available to the other. The principal has the
power of commitment, both to elicit information from the agent and to provide
signals about her own information. The principal and the agent communicate
their signals to each other, and select their actions independently based on
this communication. Each player receives a payoff based on the state and their
joint actions, and the environment moves to a new state. The interaction
continues over a finite time horizon, and both players act to optimize their
own total payoffs over the horizon. Our model encompasses as special cases
stochastic games of incomplete information and POMDPs, as well as sequential
Bayesian persuasion and mechanism design problems. We study both computation of
optimal policies and learning in our setting. While the general problems are
computationally intractable, we study algorithmic solutions under a conditional
independence assumption on the underlying state-observation distributions. We
present an polynomial-time algorithm to compute the principal's optimal policy
up to an additive approximation. Additionally, we show an efficient learning
algorithm in the case where the transition probabilities are not known
beforehand. The algorithm guarantees sublinear regret for both players
Robust Stackelberg Equilibria
This paper provides a systematic study of the robust Stackelberg equilibrium
(RSE), which naturally generalizes the widely adopted solution concept of the
strong Stackelberg equilibrium (SSE). The RSE accounts for any possible
up-to- suboptimal follower responses in Stackelberg games and is
adopted to improve the robustness of the leader's strategy. While a few
variants of robust Stackelberg equilibrium have been considered in previous
literature, the RSE solution concept we consider is importantly different -- in
some sense, it relaxes previously studied robust Stackelberg strategies and is
applicable to much broader sources of uncertainties.
We provide a thorough investigation of several fundamental properties of RSE,
including its utility guarantees, algorithmics, and learnability. We first show
that the RSE we defined always exists and thus is well-defined. Then we
characterize how the leader's utility in RSE changes with the robustness level
considered. On the algorithmic side, we show that, in sharp contrast to the
tractability of computing an SSE, it is NP-hard to obtain a fully polynomial
approximation scheme (FPTAS) for any constant robustness level. Nevertheless,
we develop a quasi-polynomial approximation scheme (QPTAS) for RSE. Finally, we
examine the learnability of the RSE in a natural learning scenario, where both
players' utilities are not known in advance, and provide almost tight sample
complexity results on learning the RSE. As a corollary of this result, we also
obtain an algorithm for learning SSE, which strictly improves a key result of
Bai et al. in terms of both utility guarantee and computational efficiency
Swap Stability in Schelling Games on Graphs
We study a recently introduced class of strategic games thatis motivated by and generalizes Schelling鈥檚 well-known resi-dential segregation model. These games are played on undi-rected graphs, with the set of agents partitioned into multi-ple types; each agent either occupies a node of the graph andnever moves away or aims to maximize the fraction of herneighbors who are of her own type. We consider a variant ofthis model that we call swap Schelling games, where the num-ber of agents is equal to the number of nodes of the graph, andagents mayswappositions with other agents to increase theirutility. We study the existence, computational complexity andquality of equilibrium assignments in these games, both froma social welfare perspective and from a diversity perspective
Markov Decision Processes with Time-Varying Geometric Discounting
Canonical models of Markov decision processes (MDPs) usually consider
geometric discounting based on a constant discount factor. While this standard
modeling approach has led to many elegant results, some recent studies indicate
the necessity of modeling time-varying discounting in certain applications.
This paper studies a model of infinite-horizon MDPs with time-varying discount
factors. We take a game-theoretic perspective -- whereby each time step is
treated as an independent decision maker with their own (fixed) discount factor
-- and we study the subgame perfect equilibrium (SPE) of the resulting game as
well as the related algorithmic problems. We present a constructive proof of
the existence of an SPE and demonstrate the EXPTIME-hardness of computing an
SPE. We also turn to the approximate notion of -SPE and show that an
-SPE exists under milder assumptions. An algorithm is presented to
compute an -SPE, of which an upper bound of the time complexity, as a
function of the convergence property of the time-varying discount factor, is
provided.Comment: 24 pages, 3 figure
Burgverein G枚sting
Blick zur Burgruine G枚stin