With this post, we start looking at player fantasy value in a different way.

Up to now, we have been looking at player fantasy value by looking at actual player performances, and comparing their fantasy points with those of their notional replacements.

Starting today, we consider a different approach as well. We will ask: if fantasy teams had a budget of fantasy points, which they had to bid to acquire players, how many fantasy points *ought* a team to spend for a player with a given production profile?

This is admittedly far removed from the everyday fantasy evaluations that fantasy owners do. But if this blog’s theme is correct, and the ideal way to evaluate fantasy players is in fantasy points, then this approach forms the basis for proper player evaluation. We have to start with the abstract and hypothetical in order to see the principles that can later be used for real world player evaluation, once the techniques have been developed.

Here are the basic assumptions I am making: each team starts off with a certain number of fantasy points, which they can spend to buy players or use as points for the season. Winning the league is based on season-long scoring, not wins in head-to-head matches. Players are acquired via auction, with fantasy points as the currency.

For example, if each team has a budget of 200 fantasy points, a team can decline to buy any players, in which case it can use its entire budget as its points on the season. If the team acquires a bunch of players via auction, it gets points in the season based on the performance of its players, and can add to its season total any points left over after paying for the players.

Using game theory, we can determine a player’s value in fantasy points as a function of team budgets and the projected fantasy point production of each player in the league.

In today’s post, we start with extremely simple assumptions in order to try to distill the most general principles of player value. Our league has just two teams, and there are only two players available. Each team starts one player, and there is one game in the season. Moreover, we know in advance how many points each player will score. Player 1 will score 20 points, and player 2 will score 10 points. Each team has a budget of 20 fantasy points to spend.

Under these particular terms, player 1 is worth 15 fantasy points, and player 2 is worth 5 fantasy points.

Note that the difference between the players’ values is equal to the difference between their point outputs: player 1 scores ten points more than player 2, and his value is also ten points higher. This seems like an obvious feature of a model where we know the player outputs. In addition, the sum of the players’ values equals each team’s budget. This is less obvious, but it is a feature, not an accident, as we will see.

We can confirm that these are the players’ values by showing that whenever two teams are bidding on the two players with the aim of maximizing their points, these bids will be the winning strategies for the players. We can use game theory to show that a player’s value is V, if we can show that if a team bids more than V, or permits another team to bid less than V, it hands the other team a winning strategy.

##### Assumption: Player 1 Is Acquired for More Than 15 Points

Recall our assumptions:

- There are two teams, A and B, playing one game
- Each team starts one player
- Each team has 20 fantasy points to spend
- Player 1 will score 20 points
- Player 2 will score 10 points
- Everyone knows how many points each player will score (and knows that everyone knows how many points each player will score, knows that everyone knows this, etc.)

Here is how we reach our player valuations. Suppose a team bids higher than 15 points for player 1. For example, suppose that team A bids 16 points for player 1. That means that team A’s total points on the season will be 24. We get this number by taking Team A’s initial budget of 20 points, subtracting 16 for the cost of player 1, and adding 20 for player 1’s fantasy production. 20 – 16 + 20 = 24.

This guarantees a winning strategy for team B. B can bid 5 fantasy points for player 2. If A does nothing, B ends up with 20 – 5 + 10 = 25 fantasy points on the season, and wins. If A matches or exceeds B’s bid, then B ends up getting no players and spending no fantasy points, ending up with 20 points – B’s original budget. But by bidding 5 for player 2, Team A ends the season with 19 points (its 24 points after bidding for player 1, minus 5 for bidding on player 2). Remember that a team can only start one player, so A will start player 1 and get his 20 points, but it gets no points for player 2’s performance, who will be on the bench. Team A bids on player 2 only to deny B the 10 points that B produces. In order to deny team B the 10 points, team A has to waste 5 points on a player that it will not start. Team A’s points will therefore be 20 (original budget) – 16 (cost of Player 1) – 5 (cost of player 2) + 20 (fantasy production of player 1) = 19. With 19 points, team A loses to team B, which retains its 20-point original budget.

I used the price of 16 for player 1 as an example, but it should be evident that the same logic applies to any bid above 15, whether bids are restricted to whole points, or permit fractional points. Anything over 15 points gives B a win by bidding 5 points for player 2.

That means we know that the highest price any team should pay for player 1 is 15. Could the price be lower than 15? The answer is no.

##### Assumption: Player 1 Is Acquired for Fewer Than 15 Points

Suppose team A bids less than 15 points for player 1, let’s say 14. Team B has the option of bidding the price up to 15, or not. If B does not, then A buys player 1 for 14 points and gets 20 – 14 + 20 = 26 fantasy points on the season. This guarantees team A a winning strategy, as follows:

Suppose team B does not bid higher than A’s bid of 14. To match team A and get to 26 points, team B needs to acquire player 2 and his 10 points for 4 points: 20 – 4 + 10 = 26. But if team B bids 4 points, team A can win by outbidding team B. A can bid 5 points. If B does not match the bid, then A gets 20 – 14 (for player 1) – 5 (for player 2) + 20 = 21. Team B acquires no players and remains at 20, losing to team A. If team B does match team A’s bid at 5, team B ends up with 25 points on the season, losing to team A at 26.

Therefore, if player 1 is acquired for 14 points, the team acquiring him is guaranteed a win as long as it follows the strategy in the last paragraph. It follows that a team cannot let the other get player A for 14 points. As before, there is nothing magical about the number 14: the same logic holds true of any bid under 15. It follows that no team will let the other team acquire player 1 for less than 15 points, because doing so will give the team acquiring player 1 a winning strategy. The players will therefore push the bidding on player 1 to 15.

Since bidding more than 15 points on player 1 is a losing strategy, and allowing the other team to bid less than 15 points for player 1 is a losing strategy, 15 points is the only rational price for player 1.

##### Player 2’s Value Must be 5 Points

If the price of player 1 is 15 points, the price of player 2 must be 5. The team that gets player 1 will get 25 points (20 – 15 + 20) on the season. Say this is team A. Team B can get 25 points by bidding 5 (20 – 15 + 10). As we did with player 1, we can show that team B bidding more than 5 points gives team A a winning strategy, and letting team A bid less than 5 also gives team A a wining strategy. Bidding exactly 5, or letting team A bid exactly 5, gives team B a tie with team A, making 5 points that rational price for player 2.

Suppose B bids more than 5 points for player 2, say 6 points. Then its season total will be 24 points = 20 (starting budget) – 6 (cost of player 2) + 10 (player 2’s points). Since team A has 25 points, team B loses.

Suppose B bids less than 5, say 4 points. It then stands to get 26 points on the season, and beat team A’s 25 points, if team A doesn’t bid player 2 up. It follows that team A will raise the bidding to 5 points. If A wins the auction at 5 points, then both teams end up with 20 points on the season: team B because it won no bids and scores no points, remaining with its original 20 point budget; and A because it spent its entire budget (giving up 15 points for player 1 and 5 for player 2), and gets 20 points of production by starting player 1.

If team B wins the auction for player 2 at 5 points, then both teams end up with 25 points. A starts with 20 points, spends 15 on player 1 and gets 20 points of production. B starts with 20 points, spends 5 on player 2 and gets 10 points of production.

Therefore, whichever team gets player 2 for 5 points, the result is a tie.

##### Generalizing the Result

This completes the proof that player 1’s value is 15 and player 2’s is 5. *Quod erat demonstrandum*, as they say. But the result is general. Whenever you have two teams with the same budget B and two players, you can determine the values of players 1 and 2 (v1 and v2) based on their points (p1 and p2) by satisfying the following:

v1 – v2 = p1 – p2

v1 + v2 = B

So if the budget is 30, and player 1 scores 10 points more than player 2, player 1’s value is 20 and player 2’s is 10, regardless of whether the numbers are 20 and 10, 25 and 15, or some other pair of numbers with a difference of 10. Raise the budget to 40 and keep the difference of 10 points, and the values become 25 and 15.

This generalization excludes absurd situations, such as negative numbers and players whose cost is greater than their point production. Negative fantasy points are possible, but no fantasy team would bid on a player which it knows will have negative point production. Similarly, no fantasy team would bid on a player whose point production is lower than his cost.

Where does this lead? Obviously the results we have obtained so far are not actionable. This is step one of a project to evaluate player fantasy values for real life fantasy sports situations. In real life, player production is not known in advance, and fantasy teams are composed of multiple players in different positions, filling different slots in starting lineups and sometimes sitting on the bench, with points production is spread out over a season or, in the case of dynasty leagues, over a career. The approach begun here is, however, one that will allow for a meaningful evaluation of players once it is refined.