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Routing without regret: On convergence to nash equilibria of regretminimizing algorithms in routing games
 In PODC
, 2006
"... Abstract There has been substantial work developing simple, efficient noregret algorithms for a wideclass of repeated decisionmaking problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversariallychanging envi ..."
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Abstract There has been substantial work developing simple, efficient noregret algorithms for a wideclass of repeated decisionmaking problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversariallychanging environments. There has also been substantial work on analyzing properties of Nash equilibria in routing games. In this paper, we consider the question: if each player in a routing game uses a noregret strategy, will behavior converge to a Nash equilibrium? In general games the answer to this question is known to be no in a strong sense, but routing games havesubstantially more structure. In this paper we show that in the Wardrop setting of multicommodity flow and infinitesimalagents, behavior will approach Nash equilibrium (formally, on most days, the cost of the flow will be close to the cost of the cheapest paths possible given that flow) at a rate that dependspolynomially on the players ' regret bounds and the maximum slope of any latency function. We also show that priceofanarchy results may be applied to these approximate equilibria, and alsoconsider the finitesize (noninfinitesimal) loadbalancing model of Azar [2].
Online Algorithms for Market Clearing
, 2002
"... In this paper we study the problem of online market clearing where there is one commodity in the market being bought and sold by multiple buyers and sellers whose bids arrive and expire at different times. The auctioneer is faced with an online clearing problem of deciding which buy and sell bids to ..."
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Cited by 46 (5 self)
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In this paper we study the problem of online market clearing where there is one commodity in the market being bought and sold by multiple buyers and sellers whose bids arrive and expire at different times. The auctioneer is faced with an online clearing problem of deciding which buy and sell bids to match without knowing what bids will arrive in the future. For maximizing profit, we present a (randomized) online algorithm with a competitive ratio of ln(p max min )+1, when bids are in a range [p min ,p max ], which we show is the best possible. A simpler algorithm has a ratio twice this, and can be used even if expiration times are not known. For maximizing the number of trades, we present a simple greedy algorithm that achieves a factor of 2 competitive ratio if no moneylosing trades are allowed. Interestingly, we show that if the online algorithm is allowed to subsidize matches  match moneylosing pairs if it has already collected enough money from previous pairs to pay for them  then it can be 1competitive with respect to the optimal offline algorithm that is not allowed subsidy. That is, the ability to subsidize is at least as valuable as knowing the future. We also consider the objectives of maximizing buy or sell volume, and present algorithms that achieve a competitive ratio of 2(ln(p max /p min ) + 1), or ln(p max /p min ) + 1 if the online algorithm is allowed subsidization. We show the latter is the best possible competitive ratio for this setting. For social welfare maximization we also obtain an optimal competitive ratio, which is below ln(p max /p min ). We present all of these results as corollaries of theorems on online matching in an incomplete interval graph.
Adaptive bound optimization for online convex optimization (extended version
, 2010
"... We introduce a new online convex optimization algorithm that adaptively chooses its regularization function based on the loss functions observed so far. This is in contrast to previous algorithms that use a fixed regularization function such as L2squared, and modify it only via a single timedepend ..."
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We introduce a new online convex optimization algorithm that adaptively chooses its regularization function based on the loss functions observed so far. This is in contrast to previous algorithms that use a fixed regularization function such as L2squared, and modify it only via a single timedependent parameter. Our algorithm’s regret bounds are worstcase optimal, and for certain realistic classes of loss functions they are much better than existing bounds. These bounds are problemdependent, which means they can exploit the structure of the actual problem instance. Critically, however, our algorithm does not need to know this structure in advance. Rather, we prove competitive guarantees that show the algorithm provides a bound within a constant factor of the best possible bound (of a certain functional form) in hindsight. 1
On learning algorithms for nash equilibria
 Algorithmic Game Theory, volume 6386 of Lecture Notes in Computer Science
, 2010
"... Abstract. Can learning algorithms find a Nash equilibrium? This is a natural question for several reasons. Learning algorithms resemble the behavior of players in many naturally arising games, and thus results on the convergence or nonconvergence properties of such dynamics may inform our understan ..."
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Abstract. Can learning algorithms find a Nash equilibrium? This is a natural question for several reasons. Learning algorithms resemble the behavior of players in many naturally arising games, and thus results on the convergence or nonconvergence properties of such dynamics may inform our understanding of the applicability of Nash equilibria as a plausible solution concept in some settings. A second reason for asking this question is in the hope of being able to prove an impossibility result, not dependent on complexity assumptions, for computing Nash equilibria via a restricted class of reasonable algorithms. In this work, we begin to answer this question by considering the dynamics of the standard multiplicative weights update learning algorithms (which are known to converge to a Nash equilibrium for zerosum games). We revisit a 3 × 3 game defined by Shapley [10] in the 1950s in order to establish that fictitious play does not converge in general games. For this simple game, we show via a potential function argument that in a variety of settings the multiplicative updates algorithm impressively fails to find the unique Nash equilibrium, in that the cumulative distributions of players produced by learning dynamics actually drift away from the equilibrium.
The Effectiveness of Opponent Modelling in a Small Imperfect Information Game
, 2006
"... Opponent modelling is an important issue in games programming today. Programs which
do not perform opponent modelling are unlikely to take full advantage of the mistakes
made by an opponent. Additionally, programs which do not adapt over time become less
of a challenge to players, causing these pla ..."
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Cited by 6 (1 self)
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Opponent modelling is an important issue in games programming today. Programs which
do not perform opponent modelling are unlikely to take full advantage of the mistakes
made by an opponent. Additionally, programs which do not adapt over time become less
of a challenge to players, causing these players to lose interest. While opponent modelling
can be a difficult challenge in perfect information games, where the full state of the game
is known to all players at all times, it becomes an even more difficult task in games of
imperfect information, where players are not always able to observe the actual state of
the game. This thesis studies the problem of opponent modelling in Kuhn Poker, a small
imperfect information game that contains several properties that make realworld poker
games interesting. Two basic types of opponent modelling are studied, explicit modelling
and implicit modelling, and their effectiveness is compared.
unknown title
"... Arguably, the birth of the Internet, the first multinetwork communications experiment, took place in November of 1977. This experiment connected the Advanced Research Projects Agency Network (ARPANET) to external sites. Protocols used included TCP (for packet switched routing) and Ethernet (to acce ..."
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Arguably, the birth of the Internet, the first multinetwork communications experiment, took place in November of 1977. This experiment connected the Advanced Research Projects Agency Network (ARPANET) to external sites. Protocols used included TCP (for packet switched routing) and Ethernet (to access a shared resource). These protocols form the backbone of computing today. E.g., without TCP and without Ethernet, we would have no Internet and no WiFi. One implicit assumption remains unchanged from 1977 to this very day. This is the belief that everyone will follow protocol, and that selfish users will not try to manipulate these protocols. This assumption makes sense if all users share a common goal, and seek to collaborate with one another towards this goal. While this may be true for the US Defense establishment in 1977, the shared common goal becomes patently absurd when you consider that 32 bit IP address no longer suffice to encompass the number of hosts on the Internet. Furthermore, it is a surprisingly simple matter to manipulate protocols such as TCP and Aloha, for gain, at the expense of others. In this thesis, we study the use of communications protocols that do not require the