Modified Knaster s Sealed Bids Approaches for Fantasy Sports Drafts

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1 Abstract Modified Knaster s Sealed Bids Approaches for Fantasy Sports Drafts Phil Poletti, Joseph Massey {ppoletti, jmassey}@wustl.edu Repo: fdfantasysports Department of Computer Science, Washington University St. Louis, MO 63130, USA December 14, 2012 In this paper, we describe two modified Knaster s Sealed Bids procedures to facilitate a fair player draft. One such procedure is a round based player allocation draft with caps on the amount a bidder can bid each item in a round. An additional approach auctions slots for a draft with point allocations carrying over in between rounds. 1. Introduction Fantasy football drafts are normally done in a rounds based draft. Drafts can be based solely on an alternation scheme or an auction based scheme with a capped amount of money to allocate for an entire team. An alternation draft methodology used for fantasy football is commonly called a serpentine or snake draft. Bidders are assigned a position from 1 to the number of teams which is commonly 12. The lowest performing teams from the previous season get the highest position in the draft. For each round there are pick numbers within the round up to the number of teams. The pick breakdown for the teams in the draft would look like the following figure. Each row shows the positions for a particular round. TEAMS ROUND Table 1: Serpentine Draft The picks can be extrapolated for the 12 bids and 12 rounds using the same scheme. In the snake draft the first bidder to choose has an advantage over subsequent positions in the draft. Another draft type used for fantasy football is an auction draft. An auction draft is used to provide a fair market approach to player allocation. Each round consists of an auctioneer putting a player on the auction block. There is an open bid process and the person bidding the most for the player wins that player. The other player s money returns to them and their bid is not taken into account to provide some means of ensuring fairness. The only fairness in the system is by asserting a cap that a bidder can spend on their entire team. 1

2 According to the rules used for the ESPN draft, a team consists of at least 16 players of which there are 9 starters. The team can be broken down into the following. POSITIONS STARTERS MAXIMUM Quarterback (QB) 1 4 Running Back (RB) 2 8 Running Back/Wide Receiver (RB/WR) 1 N/A Wide Receiver (WR) 2 8 Tight End (TE) 1 3 Team Defense/Special Teams (D/ST) 1 3 Place Kicker (K) 1 3 Bench (BE) 7 N/A For the purposes of this paper, a team consisting of the following will be generated. A team which is comprised of the starters will be allocated. POSITIONS STARTERS Quarterback (QB) 1 Running Back (RB) 2 Wide Receiver (WR) 3 Tight End (TE) 1 Team Defense/Special Teams (D/ST) 1 Place Kicker (K) 1 Furthermore, both of the approaches outlined in these procedures either implement or seed a draft in which a player of a certain position type is drafted for a round. This differs from an actual draft in either the snake or auction draft where a bidder can choose a player of any position type for any turn of the draft. 2. Modified Approaches 2.1 Sealed Bid Player Allocation Approach Knaster s sealed bid procedure allows for a number of items to be distributed among a number of people. Fairness is ensured by using the amount of money that the people bid upon the items to ensure that everyone gets their fair share through an allocation of money or points in addition to the discrete item allocation. The modified procedure for player allocations defined in this paper outlines an approach which sets a maximum bid when bidding on individual players which are the items for this implementation. This approach differs from that of Knaster's sealed bid procedure which allows bidders to bid as much as they want on any item. Each round is for a particular position type meaning that the pool of players to choose from and place bids on consists solely of one position type. A round consists of bidders declaring what they think to be their value of the players in the player pool up to the value they have to spend for an item. B1 B2 B3 B4 B5 FOSTER PERTERSO N MARTIN RICHARDSON RICE LYNCH BRADSHAW TOTAL VAL ITEMS REC Foster, Peterson Martin, Bradshaw Rice Richardson Lynch, VAL REC IFS DIFF SOS AFS FINAL SET (RND) Table 2: Player Allocation Sample Output 2.2 Slot Allocation Approach A slot allocation approach was also implemented to show how Knaster s sealed bids can be used to define the alternation scheme for an auction. This implementation instead of allocating players allocates slots in the draft. The bidder which wins the slot chooses first with the bidder bidding the second most getting the second player allocation for that slot and so on. The auction process thereby sets the draft order for the bidders in the draft. An upside to using this approach is that the player makeup of a team can be controlled by having bidders bid on slots for a particular position type. In this approach certain position types could have multiple slots if more players of that position type are required. An example would be the following for the running back position. Bidder 1 Bidder 2 Bidder 3 Slot Slot Table 3: Position Allocation In this example there are two slots to be allocated. For the first slot bidder 1 wins the first allocation for that slot and bidder 2 wins the second allocation. Finally bidder 3 wins the third allocation. Bidder 1 did not value slot 2 as highly and therefore received the second allocation for the slot with bidder 2 getting the third allocation and bidder 3 winning the first allocation. This would see a draft consisting of the following order. 2

3 DRAFT # BIDDER Table 4: Running back draft In addition to slot allocation for one position type, a slot allocation of a mixed category can be defined. One instance could be in the case where a mixed running back and quarterback slot could be auctioned. The results of the first round have a bearing on subsequent rounds in the slot allocation procedure. The points that are won or lost in the first round are used to define a higher or lower cap on the next round. For example if a bidder is owed 1 dollar from the first round the max bid per item is increased by 1/n in the next round where n is the number of items in the next round. In terms of the simulation defined for the slot allocation procedure, this increases or decreases the upper bound of the range of which the utility for an item is chosen from. 3. Implementation 3.1 Player Allocation Approach The implementation is written in python and is a modification of Knaster s sealed bids approach in terms of a cap on the maximum bid per item. A preferences file is used to capture the utility that bidders have for the pools of players. The preferences file is randomly generated by declaring the utility that each bidder has for each player. The utility is randomly chosen for each player on the interval [1,maxBid]. bid is the defined to be the maximum allocation a bidder can have for one discrete item. The preference file is then passed to Knaster s sealed bid procedure which computes who wins each item as well as the initial fare share, difference, share of the surplus, adjusted fair share, and final settlement with relation to each bidder. The parameters to the algorithm are maxbid which is the max number that a bidder can bid on an item and numbidders which represents the number of bidders/teams in the draft. The players to be allocated are populated in a template file for each position type. The following is a sample entry. A colon separates the category (position type) from the players which are comma delimited. Quarterback: Rodgers, Manning, Griffin_III Running Back: Foster, Peterson, Martin, Richardson 3.2 Slot Allocation Approach The slot allocation procedure uses the same template definition but in this case the items are slots in the draft. The following entries are for the template file for the slot allocation procedure. These entries define one slot for the quarterback position and two slots for the running back position. Quarterback: Slot1 Running Back: Slot1, Slot2 One major addition is the carryover of points from one round to the next. This is done by adjusting the original preference structure to increase or decrease the upper bound of the max bid for the next round as a function of the number of slots in the next round (where n is the current round). maxbid(n + 1) bidder = final settlement bidder (n) number of slots in next round(n) The calculations for the new preferences are done in between rounds with modifications to the preference structure to account for the previous rounds final settlement as a function of the previous round s final settlement. The output of this procedure produces the draft order for the entire draft with the constraint that a team must choose a player of the same category as that of the slot for which it has been allocated. This can be extended to mixed categories. 4. Results and Analysis 4.1 Player Allocation Approach The results can be discussed in terms of the key fair division criteria. Proportionality The player allocation is proportional due to the proportions for each team being more than 1/n for n is the number of teams. The minimum proportions for the 12 team allocation yielded around 13% which is greater than the 8.33% needed for proportionality. For smaller leagues the proportionality was larger as can be seen from the case for 5 bidders. 3

4 Proportionality Proportionality Player Allocation 34.00% 33.50% 33.00% 32.50% 32.00% 31.50% 31.00% 30.50% 30.00% Run Figure 1: Proportionality for 200 runs (5 bidders) PROPORTIONALITY BIDDER Min Average % 14.99% 14.41% % 15.04% 14.40% % 14.98% 14.39% % 15.21% 14.39% % 14.98% 14.38% % 15.14% 14.40% % 15.16% 14.38% % 14.95% 14.39% % 14.92% 14.39% % 15.06% 14.41% % 14.92% 14.37% % 15.09% 14.40% Table 5: Proportionality for 200 runs (12 bidders) Envy-freeness There may be envy in the allocation due to the fact that some bidders receive a larger share than others. For example, suppose there are 3 bidders, A, B, C trying to divide a single item and bidder A values the item the most. If bidder B values that item more that bidder C, bidder B will be compensated more, thus making bidder C envious of bidder B. Therefore, as long as the number of items is less than the number of bidders minus 1, this algorithm is envy free. In other words, as long as all but one bidders receive an item, envy-free is satisfied but this is not guaranteed. Efficiency The player allocation approach is efficient due to the fact that it allocates all of the players in the pool. The issue is that allocating all of the players in the pool would allocate more than the allowed players for certain positions. Points Settlement For the player allocation approach the final settlement is a running total of the settlements for each round. In this procedure the settlement can become large in the negative or positive direction due to settlements not carrying over in between rounds. As the number of bidders increased the absolute values of the final settlements decreased. This is due to more variability of the preference for more bidder/teams in the system. Figure 2 and Table 6 show the system scaling for an increase in the number of bidders. Equitability The play allocation approach is not equitable due to teams not receiving equal allocations. 4

5 Final Settlement 200 Final Settlement Player Allocation 4.2 Slot Allocation Approach Proportionality In the slot allocation approach bidders get what they deem to be more than their fair share of the slots. The proportionality for a smaller bidder sizes is proportional with all values being greater than 27% which is larger than the 20% needed to satisfy proportionality Proportionality Slot Allocation % % % % Figure 2: Final settlement 100 runs (5 bidders) BIDDER POINTS SETTLEMENT Negative Run Positive Table 6: Negative and Positive Settlements for 100 runs 35.00% 30.00% 25.00% 20.00% Figure 3: Proportionality for 200 runs (5 bidders) The procedure also scales to 12 bidders which is the normal amount of teams in a fantasy football league. The proportionality is around 13% for 12 bidders. For proportionality to be satisfied for 12 bidders only roughly 8.33% needs to be allocated to each bidder. So proportionality for the slots is satisfied. As the number of bidders increases, the variability of the proportionality decreases. 5

6 Final Settlement PROPORTIONALITY BIDDER Min Average % 28.67% 16.83% % 26.24% 16.91% % 22.50% 16.87% % 24.17% 16.82% % 25.57% 17.27% % 27.50% 16.99% % 33.91% 16.98% % 25.90% 16.87% % 22.38% 16.89% % 24.31% 16.82% % 25.71% 17.03% % 25.14% 16.94% Table 7: Proportionality for 200 runs (12 bidders) One issue with this procedure is that the proportionality is only being calculated for the first pick per slot and is not taking into account how a bidder who wins the first pick of a slot feels about the others picks for a slot are being allocated. This is in the context of who comes in second, third, and so on up to the number of teams. Envy-freeness This algorithm is not envy free because although everyone gets a proportional share or greater of the slots to be allocated some may envy another person s share. Points Settlement The point settlements show the amount of money that each bidder will receive or give to the other bidders in the procedure. Setting the constraint for 10 points maximum allocation for each item and having the point carry over into the next round keeps the settlements from being too large in either direction. For 100 runs there were no unexpected results. Values were normally close to zero. Final Settlement Slot Allocation Equitability This procedure is not equitable due to each bidder not receiving an equal share of the slots. Efficiency In terms of allocating the slots the procedure is efficient. -20 Run Figure 4: Final settlement 100 runs (5 bidders) 6

7 BIDDER POINTS SETTLEMENT Negative Positive Table 8: Negative and Positive Settlements for 100 runs 5. Conclusions 5.1 Player Allocation Approach Functionally, the Knaster s sealed bid modified procedure for player allocations can be compared to the auction based approach normally used for fantasy drafts. One downfall of the sealed bid procedure player allocation defined in this paper is the fact that if a pool of more players than bidders is used, some bidders could end up with more position allocations than the rules warrant. A scheme would need to be implemented where players could be put into a pool and points are given in compensation. Also, because of the pure nature of Knaster s sealed been approach, not all teams are guaranteed to receive any player of a certain position. 5.2 Slot Allocation Approach The slot allocation can be compared to the snake draft method discussed in the introduction. The output of the slot allocation is the order that the teams will chose the players. This particular scheme could even be used in conjunction with a database which defines the top players for each category to automate a draft based on which of the top players are available for a position. In terms of adhering to rules for what the composition of a team by allocations per position type, the slot allocation approach performs better than the player allocation approach. This is due to the fact that you can strictly enforce what types of players need to be allocated. 7