BIDDING STRATEGIES FOR FANTASY-SPORT AUCTIONS BY ANAGNOSTOPOULOS ET AL. Alireza Amani Hamedani Sepehr Mousavi
§ Introduction § Absence of Nash equilibrium § Fair-Price Bidding 2
§ Type of online game § Participants as virtual managers of professional athletes § Choosing players and modifying rosters over the course of a season § Fantasy points obtained based on statistical performance of the athletes in actual games 3
§ Fantasy Points § Converted athlete statistics from real-life games § Calculation § Manually by league commissioner § Online platforms tracking game results 4
§ Users (team managers) add, drop, and trade athletes over the course of the season § In response to changes in athletes’ potentials § Pivotal event is the player draft § Initiates the competition 5
§ Multi-billion dollar industry § In 2017, 59.3 million users in the USA and Canada § On average, fantasy sport players spend $556 over a 1-year period 6
Fig 1. Number of Fantasy Sports Users by Year (in millions) in the USA and Canada 7
§ Snake vs Auction § Snake Draft § Teams taking turns choosing players based on pre-determined order § Once each round is over, the draft snakes back on itself § Used in majority of fantasy leagues 8
Fig 2. Snake draft with 12 teams and 15 rounds 9
§ Auction Draft § Each team has an initial budget and each player has a price § The number of rounds mirrors the number of roster spots § Instead of drafting a player in your turn, you place a player on the auction block and start the bidding at an amount of your choice. § Focus of this paper 10
§ Fantasy sports league composed of: § 𝑙 team managers , or users , with 3 ≤ 𝑙 ≤ 20 § 𝑣 ( , 𝑣 * , … , 𝑣 , § 𝑜 athletes (or players ) § 𝑄 ( , 𝑄 * , … , 𝑄 / § Each team composed of 𝑛 athletes § Depends on the sport and fantasy games provider 11
§ Snake vs Auction § Choice made by the initiator of the league § Snake Draft § No bidding or competition, just a pre-determined order of teams to draft 12
§ Auction Draft § Fixed budget of 𝐶 § Managers taking turns successively, in some pre-determined order, nominating athletes for bidding via an English auction § Default bid is $1. Can be raised higher within the budget § Managers given a fixed amount of time to place higher bid 13
§ Auction Draft § Leftover money cannot be used § Managers should be able to complete their rosters § Each athlete has a fixed position and each team must meet a fixed distribution of positions § Depends on the sport and fantasy games provider 14
§ Simplifying assumptions § Team managers agree on the value of every athlete. § Each athlete 𝑄 2 has an associated value 𝑤 2 § 𝑤 2 : Expected number of fantasy points 𝑄 2 will earn throughout the season § 𝑤 2 is a shared belief, common to all managers 15
§ Simplifying assumptions § Auction draft is a sealed-bid auction § Arbitrary fractional bids § In case of a tie, athlete is given out with equal probability § Exception when all managers place the minimal bid. Athlete given to nominating manager 16
§ Simplifying assumptions § For each position the player pool has exactly the number of athletes required to complete each team § 𝑜 = 𝑙𝑛 § Fair share: / 𝑊 = 1 𝑙 7 𝑤 2 28( 17
§ Pure strategy subgame perfect Nash equilibria do not generally exist in the fantasy auction model § 9 : worst case with athletes automatically nominated in decreasing order of their values § 9 (; worst case for the general case in which nominations are made in a general adaptive fashion according to manager strategies 18
§ Due to competitive and strategic environment, it is natural to take game theoretic approach. § Generally there will not even exist any pure strategy subgame perfect equilibria in fantasy draft auctions 19
§ Example: § Two users with equal budgets § Each team roster has two slots § Four athletes, two of unequal positive value, and two of value 0 § Claim: In the above example, if the lower (positive) value athlete is nominated first, there exists no pure strategy Nash equilibrium forward from that point 20
§ Simple but not good approach § Generalized good approach § Fair-Price bidding in arbitrary nomination order 21
§ Define fair share of total value as below: § Define fair price for athlete 𝑄 2 : § 22
§ Not a good result in this case: 9((C,F>F) § 𝑤 ( = 𝑤 * = ⋯ = 𝑤 ,>( = 𝑊 1 − 𝜁 𝑏𝑜𝑒 𝑤 , = 𝑤 ,C( = … = 𝑤 ,D = ,D>,C( § The value of the final team for our manager: 23
§ § Expected value of the final team at least 9 : with 𝛽 = 1.5 § Regardless of the other managers’ bidding strategies 24
§ Valuable athletes with 𝑤 2 ≥ 9 L § Three scenarios: One valuable athlete is bought 1. No valuable athlete, at least at one point, not enough budget 2. No valuable athlete, always sufficient budget 3. 25
Expected value: 9 L 𝑞 1. At that point, the value at least 9 *L . So, Expected value: 9 *L 𝑞 N 2. Expected value: 𝑌𝑞 NN and, 3. 26
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§ Examples for other 𝛽 that lead to a result of almost 9 : § How to come up with a lower bound for any 𝛽 ? 28
§ Three parameters: 𝛾 > 𝛽 ≥ 1 𝑏𝑜𝑒 𝛿 ≥ 1 § Two groups: 29
§ Case 1: Value of group 𝑀 is larger than 𝑇 § Put all the budget for group 𝑀 30
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§ Case 2: § Average value per spot: § For available spots, choose an athlete when 𝑄 2 in 𝑀 , or 𝑄 2 in 𝑇 and 32
§ Complex proof § Under choices 𝛽 = (; : , 𝛾 = 8 𝑏𝑜𝑒 𝛿 = 2, Expected value is 9 (; 33
§ Results hold for private-value case § The big question: How to fill in the gap between our result and the best one can hope for. § May be competitive in real life! 34
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