7,152 research outputs found
All-Pay Bidding Games on Graphs
In this paper we introduce and study {\em all-pay bidding games}, a class of
two player, zero-sum games on graphs. The game proceeds as follows. We place a
token on some vertex in the graph and assign budgets to the two players. Each
turn, each player submits a sealed legal bid (non-negative and below their
remaining budget), which is deducted from their budget and the highest bidder
moves the token onto an adjacent vertex. The game ends once a sink is reached,
and \PO pays \PT the outcome that is associated with the sink. The players
attempt to maximize their expected outcome. Our games model settings where
effort (of no inherent value) needs to be invested in an ongoing and stateful
manner. On the negative side, we show that even in simple games on DAGs,
optimal strategies may require a distribution over bids with infinite support.
A central quantity in bidding games is the {\em ratio} of the players budgets.
On the positive side, we show a simple FPTAS for DAGs, that, for each budget
ratio, outputs an approximation for the optimal strategy for that ratio. We
also implement it, show that it performs well, and suggests interesting
properties of these games. Then, given an outcome , we show an algorithm for
finding the necessary and sufficient initial ratio for guaranteeing outcome
with probability~ and a strategy ensuring such. Finally, while the general
case has not previously been studied, solving the specific game in which \PO
wins iff he wins the first two auctions, has been long stated as an open
question, which we solve.Comment: The full version of a paper published in AAAI 202
Determinacy in Discrete-Bidding Infinite-Duration Games
In two-player games on graphs, the players move a token through a graph to
produce an infinite path, which determines the winner of the game. Such games
are central in formal methods since they model the interaction between a
non-terminating system and its environment. In bidding games the players bid
for the right to move the token: in each round, the players simultaneously
submit bids, and the higher bidder moves the token and pays the other player.
Bidding games are known to have a clean and elegant mathematical structure that
relies on the ability of the players to submit arbitrarily small bids. Many
applications, however, require a fixed granularity for the bids, which can
represent, for example, the monetary value expressed in cents. We study, for
the first time, the combination of discrete-bidding and infinite-duration
games. Our most important result proves that these games form a large
determined subclass of concurrent games, where determinacy is the strong
property that there always exists exactly one player who can guarantee winning
the game. In particular, we show that, in contrast to non-discrete bidding
games, the mechanism with which tied bids are resolved plays an important role
in discrete-bidding games. We study several natural tie-breaking mechanisms and
show that, while some do not admit determinacy, most natural mechanisms imply
determinacy for every pair of initial budgets
Infinite-Duration Bidding Games
Two-player games on graphs are widely studied in formal methods as they model
the interaction between a system and its environment. The game is played by
moving a token throughout a graph to produce an infinite path. There are
several common modes to determine how the players move the token through the
graph; e.g., in turn-based games the players alternate turns in moving the
token. We study the {\em bidding} mode of moving the token, which, to the best
of our knowledge, has never been studied in infinite-duration games. The
following bidding rule was previously defined and called Richman bidding. Both
players have separate {\em budgets}, which sum up to . In each turn, a
bidding takes place: Both players submit bids simultaneously, where a bid is
legal if it does not exceed the available budget, and the higher bidder pays
his bid to the other player and moves the token. The central question studied
in bidding games is a necessary and sufficient initial budget for winning the
game: a {\em threshold} budget in a vertex is a value such that
if Player 's budget exceeds , he can win the game, and if Player 's
budget exceeds , he can win the game. Threshold budgets were previously
shown to exist in every vertex of a reachability game, which have an
interesting connection with {\em random-turn} games -- a sub-class of simple
stochastic games in which the player who moves is chosen randomly. We show the
existence of threshold budgets for a qualitative class of infinite-duration
games, namely parity games, and a quantitative class, namely mean-payoff games.
The key component of the proof is a quantitative solution to strongly-connected
mean-payoff bidding games in which we extend the connection with random-turn
games to these games, and construct explicit optimal strategies for both
players.Comment: A short version appeared in CONCUR 2017. The paper is accepted to
JAC
The Bidder's Curse
We employ a novel approach to identify overbidding in the field. We compare auction prices to fixed prices for the same item on the same webpage. In detailed board-game data, 42 percent of auctions exceed the simultaneous fixed price. The result replicates in a broad cross-section of auctions (48 percent). A small fraction of overbidders, 17 percent, suffices to generate the overbidding. The observed behavior is inconsistent with rational behavior, even allowing for uncertainty and switching costs, since also the expected auction price exceeds the fixed price. Limited attention to outside options is most consistent with our results.
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