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Game Theory in Baseball Salary Arbitration Hearings
Andrew Miller, Class of 2007
October 27, 2005

Introduction
Beginning in 1974, baseball players with between three and six years of service experience in Major League Baseball have had the option of filing for Final Offer Salary Arbitration at the end of each season. Due to the nature of the arbitration process, a complicated game has evolved where Players and Clubs must simultaneously submit salary offers that will maximize their payoffs through either negotiated settlement or through increased probability of winning at the arbitration hearing. The salary arbitration bidding process is a simultaneous game because even though the player and the Club may make their offers at different times, both parties submit their bids without knowing what the other side has bid. Some information is available regarding the other side’s bidding strategy from previous negotiation efforts, though the value of this information is questionable since both sides may deviate significantly from previous offers when selecting a final offer to submit in arbitration.

Rules of the Game
The rules governing salary arbitration hearings are set forth in the Collective Bargaining Agreement (“CBA”) negotiated between the 30 Major League baseball teams (the “Clubs”) and the Major League Baseball Player’s Association (“MLBPA”). The most recent agreement became effective September 30, 2002 and expires on December 19, 2006. Any eligible player may submit the issue of his salary to “final and binding arbitration” without the consent of the Clubs. Within 3 days of submitting for arbitration, the player and the Club exchange salary offers. The player may withdraw the matter within 7 days after receiving the Club’s arbitration salary offer and the player and Club may reach a negotiated settlement prior to the arbitration hearing. If the matter is not withdrawn before the hearing then a three person panel hears arguments from both parties after which it must choose either the player’s salary offer or the Club’s salary offer based on a fixed set of criteria defined in the CBA.

Payoffs
Salary arbitration hearings can result in one of two payoffs, the Player’s bid (P) and the Club’s bid (C). In an overly simplistic model of a negotiation, a Player’s true market value, M, would be set such that C<M<P. Not taking into account the rules of the CBA noted above, M would be reached in an iterative negotiation process where the player and the Club would trade offers until a settlement could be reached somewhere between each party’s opening bid (assuming a process similar to a free agent negotiation). A player without free agent eligibility would have a lower M than a comparable free agent, however for the purpose of this game, a negotiated settlement between an arbitration-eligible player and a Club is assumed to be equal to M.

Bidding Tactics
Since either party can unilaterally decide to play the game (i.e. submit for salary arbitration), subject to certain restrictions, and because the payoff in arbitration hearings must be either C or P, bidding tactics are one of the few dynamic elements of the game. The penalties that exist for non-optimal bids shape the bidding tactics. For example, if a Player bids too high, then the Club will win the hearing and the payout will be C, the Player’s worst payout. This penalty to the Player for his overbid will be M-C because he could have settled at M, had he not overbid.

If the Player underbids he would be likely to win an arbitration hearing, which would mean that the payout would be >M which is not in the best interests of the Club. Therefore, if the player underbids, the club has a high probability of negotiating to avoid arbitration, for a value of M. Therefore, the penalty to the Player for his underbid will be P-M because he could have won P had he not underbid.

Conversely, if the Club underbids, then the Player will be more likely to win the hearing and the payout will be P, the worst payout for the Club. The penalty for the Club’s underbid will be P-M because the Club could have settled for M prior to the hearing. Likewise to the above case, if the Club overbids, then the Player captures the most value by negotiating a settlement with a payout of M and avoiding arbitration. The penalty for the Club’s underbid is equal to M-C because it would have won C had it not overbid.

The table below summarizes the payouts for these potential scenarios:

Graphically the value capture can be represented as shown below:

Based on the payoffs above, both the Club and Player will attempt to make a bid that captures the maximum value between C and M or P and M respectively.

Second Stage Game
After bids have been submitted, the two parties have the ability to negotiate a settlement prior to the arbitration hearing. This negotiated settlement adds a layer of complexity to the salary arbitration game. Each party will attempt to determine their odds of winning the arbitration after other side’s bid is revealed. Based on the expected value of the potential payoffs, the two sides will either decide to allow the arbitration hearing to determine the payoff or will negotiate a settlement. This settlement will occur when the settlement payoff meets or exceeds the expected value placed of arbitration as determined by each side. It is likely that there is a greater probability of entering negotiations as P and C approach M. The decision tree below demonstrates how a Club, for example, would decide whether to settle or move forward to an arbitration hearing.

In the above example the Club will rationally agree to settle at any point below a salary of $1.3 million. The Player will most likely have a completely different decision tree which may lead to an arbitration hearing if the Player determines his expected value is much greater than $1.3 million. The question of how each party assesses the probability of the payoffs in determining the bidding and post-bidding strategy remains unanswered but would make an interesting topic of study.

Historical Data and Calculating “M’
Since 1979, there have been 391 arbitration hearings of which the Clubs have won 56%. In recent years as high as 87% of all arbitration filings have resulted in negotiated settlements (with M from our theoretical example above as the payoff) prior to a hearing.

Historically, arbitration awards can be measured in relation to players’ marginal revenue product, or the amount of revenue that a player contributes to a club, which can be measured in a basic form by finding the additional revenue from each Club victory and finding the Player’s contribution to the Club’s victories.

As years of service increase, arbitration awards increase as a percentage of marginal revenue product (see figure #2). Historical arbitration awards, whether a win for the Player or Club (payoff P or C), can be used to approximate M because this tells us the value the market (in this case arbitrators) have placed on arbitration-eligible players. Data that includes the bids and negotiated settlements for filed arbitration cases would help more accurately estimate M.

Source: Burger and Walters

This expected arbitration value of M can be calculated with the formula: EV arbitration = MRP player * Historical % MRP

For example, a Player with four years of service could calculate his MRP and find an expected value, EV arbitration = MRP player * 44% = 0.44MRP player.

However, this assumes that Players and Clubs acted optimally in historical iterations of the game. Assuming that at least one side has acted sub-optimally in past situations, then the historical information may not be as useful a tool to determine optimal bidding strategy. Instead the Club, for example, should solve the game for every possible value for P. Each P will correspond to a probability and corresponding expected value which will allow the Club to determine the optimal C to get their maximum payoff.

Using “M” in Bidding Strategy
Using the above historical information it may be possible for a Club or Player to devise an optimal bidding strategy. Let’s assume that the goal in an arbitration hearing for a Club is to limit the losses from potential under/over bidding and maximize the probability of winning a hearing. That is the Club wants to minimize the expected value of the cost of losing a hearing while maximizing the value that it captures between C and M. In doing so, the Club will attempt to maximize the expected value of both stages of the game: the payoff in settlement or the alternative payoffs at an arbitration hearing.

The bid with the highest probability of winning the hearing would be a bid where C = M. However, if the Player is rational and knows M, then a bid of C=M would guarantee either a settlement (M) or arbitration payoff (C) of that amount. Thus if the Club can bid such that C<M, and increase the expected value for the Club, it should bid that amount. For example, a bid where C = 0.90*M, may have a 95% chance of winning at arbitration. The Club would have a higher expected value by bidding this lower amount than it would by bidding at C=M, depending on the Player’s bid (see example below). In this example, the Club could save $117,500 by bidding below M.

These scenarios demonstrate that both the Club and the Player would be attempting to determine the tradeoff between probability of winning the arbitration hearing and the salary that would be awarded in that payoff. The Club would decrease C until a point is reached in this tradeoff that maximizes the expected value for the Club. Further application of auction bidding strategy could provide more detail on optimization of this bidding process.

Limitations
Determining bids using this approach does have its limitations. It assumes that M is equal to the MRP’s of historical arbitration awards. To give a more accurate estimation of M, arbitration bids and settlement values from the 87% of cases settled prior to hearing should also be analyzed.

Additionally, based on discussions I have had, using MRP may not be possible because arbitrators have interpreted the criteria outlined in the CBA as not allowing evidence of non-traditional performance related statistics. The CBA does specify, however, that the criteria for arbitration may include the “quality of the Player’s contribution to his Club during the past season (including but not limited to his overall performance, special qualities of leadership and public appeal)," suggesting that use of MRP may be legally justified during arbitration. Despite the fact that arbitrators generally reject this interpretation of the CBA, knowledge and use of MRP by Clubs and Players can still be useful in determining an optimal bidding strategy.

Conclusion
Baseball salary arbitration cases pose interesting bidding strategies that may be optimized through the study of game theory. The above paper merely scratches the surface in outlining the game theory in a very simple manner. Interesting topics for more in-depth analysis could consist of determining how each side optimizes its potential payoff, a more accurate estimate of M, how M is accounted for in determining P and C, or calculation of reaction functions based on the opposition’s potential bid. Variations in bidding strategies could be studied based on the Players (or agents) or Clubs involved (do Scott Boras’ clients play the game different than clients of other agents or do the Yankees play the game different than the Devil Rays).

Additionally, it would be interesting to determine whether bidding strategies are affected because Clubs play this game multiple times in a given season or over multiple seasons, while a Player may play this game once or twice in his career. This might result a reputation effect, where Clubs develop bidding patterns over time that Players can interpret before making bids. It might also result in a precedent effect, where the Clubs are more likely to bid conservatively or settle before hearing to avoid the damaging salary precedent created by the Player winning at arbitration.

References

Collective Bargaining Agreement Article VI, §F (1) – (4)

Arbitration Bias and Self-Interest: Lessons from the Baseball Labor Market, John D. Burger and Stephen J.K. Walters, 2005.

Baseball Arbitration After 20 Years, John Fizel, 1994

Collective Bargaining Agreement Article VI, §F (12)(a).