## Strategic Betting for Competitive Agents

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Citations: | 2 - 1 self |

### BibTeX

@MISC{Wagman_strategicbetting,

author = {Liad Wagman and Vincent Conitzer and Duke Unversity},

title = {Strategic Betting for Competitive Agents},

year = {}

}

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### Abstract

In many multiagent settings, each agent’s goal is to come out ahead of the other agents on some metric, such as the currency obtained by the agent. In such settings, it is not appropriate for an agent to try to maximize its expected score on the metric; rather, the agent should maximize its expected probability of winning. In principle, given this objective, the game can be solved using game-theoretic techniques. However, most games of interest are far too large and complex to solve exactly. To get some intuition as to what an optimal strategy in such games should look like, we introduce a simplified game that captures some of their key aspects, and solve it (and several variants) exactly. Specifically, the basic game that we study is the following: each agent i chooses a lottery over nonnegative numbers whose expectation is equal to its budget bi. The agent with the highest realized outcome wins (and agents only care about winning). We show that there is a unique symmetric equilibrium when budgets are equal. We proceed to study and solve extensions, including settings where agents must obtain a minimum outcome to win; where agents choose their budgets (at a cost); and where budgets are private information.

### Citations

1305 |
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(Show Context)
Citation Context ...t has. However, at least in the competitions described above, the predominant goal is simply to win.) There are very powerful axiomatic arguments for maximizing expected utility (for an overview, see =-=[16]-=-), and nothing in this paper conflicts with maximizing expected utility. Once the utility function is correctly defined, we can in principle solve such strategic settings using game-theoretic solution... |

262 | A Definition of Subjective Probability
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(Show Context)
Citation Context ...0 xdFi(x) = bi. Thus, a pure strategy for an agent in this game is any fair lottery over nonnegative numbers. Any mixed strategy (consisting of a distribution over lotteries—a compound lottery in the =-=[2]-=- framework) can be reduced to a pure strategy by considering its reduced lottery, the (simple) lottery that generates the same ultimate distribution over outcomes. Hence, we do not need to consider mi... |

127 | Approximating game-theoretic optimal strategies for full-scale poker
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(Show Context)
Citation Context ...ptimal strategy—although many creative approaches have been proposed to compute a strategy that is close to an optimal strategy, both for the Trading Agent Competition [21, 22, 20] and computer poker =-=[18, 5, 11, 12]-=-). We will make no attempt at solving them in this paper. Instead, we will study a much simpler game that nevertheless illustrates many of the key phenomena in these competitive settings. The most bas... |

46 | Strategic interactions in a supply chain game
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(Show Context)
Citation Context ...lve for the game-theoretically optimal strategy—although many creative approaches have been proposed to compute a strategy that is close to an optimal strategy, both for the Trading Agent Competition =-=[21, 22, 20]-=- and computer poker [18, 5, 11, 12]). We will make no attempt at solving them in this paper. Instead, we will study a much simpler game that nevertheless illustrates many of the key phenomena in these... |

45 | A competitive texas hold’em poker player via automated abstraction and real-time equilibrium computation
- Gilpin, Sandholm
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(Show Context)
Citation Context ...ptimal strategy—although many creative approaches have been proposed to compute a strategy that is close to an optimal strategy, both for the Trading Agent Competition [21, 22, 20] and computer poker =-=[18, 5, 11, 12]-=-). We will make no attempt at solving them in this paper. Instead, we will study a much simpler game that nevertheless illustrates many of the key phenomena in these competitive settings. The most bas... |

39 | The Penn-Lehman Automated Trading Project
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(Show Context)
Citation Context ...agent with the most money in the end. Yet another example is the Penn-Lehman Automated Trading (PLAT) live competition, in which automated stock trading agents compete based on real stock market data =-=[13]-=-. (Unlike the other competitions, this one is no longer active, though apparently not due to lack of interest.) In games such ∗ Communicating author. Tel: (919) 323-6647, Fax: (919) 684-8974 1sas thes... |

35 |
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Citation Context ...ptimal strategy—although many creative approaches have been proposed to compute a strategy that is close to an optimal strategy, both for the Trading Agent Competition [21, 22, 20] and computer poker =-=[18, 5, 11, 12]-=-). We will make no attempt at solving them in this paper. Instead, we will study a much simpler game that nevertheless illustrates many of the key phenomena in these competitive settings. The most bas... |

34 |
Patent Races and Optimal Patent Breadth and Length
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(Show Context)
Citation Context ...ation, and the patent is awarded to the agent with the greatest innovation; however, here, there is typically also a minimum level of innovation that needs to be reached in order to obtain the patent =-=[8, 9]-=-. (Later in the paper, we will study the variant of our game where agents must obtain at least a certain value to win.) Other applications include political campaigns and arms races. In a working pape... |

29 |
Optimal auction with financially constrained buyers
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- 1996
(Show Context)
Citation Context ...in. For an overview on all-pay auctions, see [3]. “Access to a fair insurance market” means that agents can place any fair bet.) Dulleck et al. are motivated in part by a result by Laffont and Robert =-=[15]-=-, who study the optimal (revenue maximizing) auction when bidders face (common knowledge) financial constraints. Laffont and Robert show that the optimal auction in this case takes the form of an all-... |

29 |
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(Show Context)
Citation Context ...lve for the game-theoretically optimal strategy—although many creative approaches have been proposed to compute a strategy that is close to an optimal strategy, both for the Trading Agent Competition =-=[21, 22, 20]-=- and computer poker [18, 5, 11, 12]). We will make no attempt at solving them in this paper. Instead, we will study a much simpler game that nevertheless illustrates many of the key phenomena in these... |

28 | An Analysis of the 2004 Supply Chain Management Trading Agent Competition
- Kiekintveld, Vorobeychik, et al.
- 2005
(Show Context)
Citation Context ... competition, there may be a specification that if no agent has positive profit (which does sometimes happen: for example, in the early rounds of the 2003 Supply Chain Management TAC, as described in =-=[14]-=-), then nobody wins. Also, in the patent race application from economics, a minimum level of innovation must be reached or surpassed for a patent to be granted. We wish to solve for the symmetric equi... |

24 | Better automated abstraction techniques for imperfect information games, with application to texas hold’em poker
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- 2007
(Show Context)
Citation Context |

21 |
The supply chain management game for the 2007 trading agent competition
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- 2006
(Show Context)
Citation Context ... do. Most notably, agents are often in competition with each other, and the goal is to come out ahead of the other agents. For example, in the Trading Agent Competition’s Supply Chain Management game =-=[7]-=-, computer programs make decisions about managing a (simulated) supply chain. Their tasks include negotiating supply contracts, bidding for customer orders, and managing assembly and shipping. The win... |

18 |
Two-Stage Patent Races and Patent Policy
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(Show Context)
Citation Context ...ation, and the patent is awarded to the agent with the greatest innovation; however, here, there is typically also a minimum level of innovation that needs to be reached in order to obtain the patent =-=[8, 9]-=-. (Later in the paper, we will study the variant of our game where agents must obtain at least a certain value to win.) Other applications include political campaigns and arms races. In a working pape... |

16 |
de Vries. The all-pay auction with complete information. Economic Theory, 8:291–305
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(Show Context)
Citation Context ...t for their budget, and has access to a fair insurance market. (An all-pay auction is an auction in which each agent must pay its bid, even if it did not win. For an overview on all-pay auctions, see =-=[3]-=-. “Access to a fair insurance market” means that agents can place any fair bet.) Dulleck et al. are motivated in part by a result by Laffont and Robert [15], who study the optimal (revenue maximizing)... |

12 | Increasing dominance with no efficiency effect
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(Show Context)
Citation Context ...aking in R&D environments. Here, a choice of technology leads to a distribution over the final quality (or improvement in quality) of the product, which determines which firm will dominate the market =-=[1, 4, 6]-=-. Patent races constitute another application, where again the choice of technology leads to a distribution over the level of innovation, and the patent is awarded to the agent with the greatest innov... |

11 | Empirical game-theoretic analysis of the TAC market games
- Wellman, Jordan, et al.
- 2006
(Show Context)
Citation Context ...lve for the game-theoretically optimal strategy—although many creative approaches have been proposed to compute a strategy that is close to an optimal strategy, both for the Trading Agent Competition =-=[21, 22, 20]-=- and computer poker [18, 5, 11, 12]). We will make no attempt at solving them in this paper. Instead, we will study a much simpler game that nevertheless illustrates many of the key phenomena in these... |

10 |
Portfolio Choice in Research and Development
- Bhattacharya, Mookherjee
- 1986
(Show Context)
Citation Context ...aking in R&D environments. Here, a choice of technology leads to a distribution over the final quality (or improvement in quality) of the product, which determines which firm will dominate the market =-=[1, 4, 6]-=-. Patent races constitute another application, where again the choice of technology leads to a distribution over the level of innovation, and the patent is awarded to the agent with the greatest innov... |

9 | Thresholded rewards: Acting optimally in timed, zero-sum games
- McMillen, Veloso
- 2007
(Show Context)
Citation Context ...ount of money, but in fact, the only thing that matters is whether the agent made more money than the other agents. More recently, agent designers have started to take this into account (for example, =-=[17]-=-). As a simple numerical example, suppose that agent 2 will certainly end up with $50, and agent 1 has a choice between two strategies. Strategy 1 will give agent 1 $40 with probability 100%; strategy... |

7 | 2007): “Go for broke or play it safe? Dynamic competition with choice of variance
- Anderson, Cabral
(Show Context)
Citation Context ...aking in R&D environments. Here, a choice of technology leads to a distribution over the final quality (or improvement in quality) of the product, which determines which firm will dominate the market =-=[1, 4, 6]-=-. Patent races constitute another application, where again the choice of technology leads to a distribution over the level of innovation, and the patent is awarded to the agent with the greatest innov... |

4 | All-pay Auctions with Budget Constraints and Fair Insurance
- DULLECK, FRIJTERS, et al.
- 2006
(Show Context)
Citation Context ...per, we will study the variant of our game where agents must obtain at least a certain value to win.) Other applications include political campaigns and arms races. In a working paper, Dulleck et al. =-=[10]-=- (independently) propose what is effectively the same game as the basic setting that we initially study in this paper, in a different context. They study all-pay auctions in which each bidder is budge... |