Solving Baseball Arbitration: Game Theory and a Statistical Model

In Major League Baseball, arbitration is a process used to determine the salary of young players that are still under team control. Most simply, the baseball arbitration process uses an independent arbiter to ensure that both a player and his team get a fair shake in a contract’s value. At its heart, baseball arbitration is more useful as a threat than as a tool. Undergoing the arbitration process is a risk for both the baseball player and for the team: the team may be required to pay more than they want or the player may get paid much less than they want. Because of this, most of the time players and teams find a deal without having to use the arbitration process.

But what if they don’t? If you are a player you obviously want to maximize your salary. If you are a team owner or general manager, you want to minimize the amount the player is paid. This introduces a natural adversarial matchup between a player and his team when entering the arbitration process. In this article we will use some basic ideas from game theory and statistics to analyze the strategies that teams and players should use when entering the arbitration process.

As is typical in mathematics, we will start with a very simple approximation of reality then add in more and more complexity until our situation looks close to reality.

Our conclusions: Players should generally avoid the arbitration process because it can be very difficult for them to get a fair valuation. In a mildly idealized setting, we will show that for a very specific choice of bid, even for a fair arbiter a player will be underpaid on average. To put it slightly differently: If ownership knows what they are doing, they can bend the baseball arbitration process so that they receive favorable verdicts to the detriment of their players.

What is Baseball Arbitration?

In a legal sense, baseball arbitration is the process of consulting an impartial third party to resolve some dispute. In baseball arbitration is used to determine the value of a young (3-6 years of being a professional) player before they are eligible for free agency. For full details, see the MLB’s article explaining the process here.

What I love about mathematical modelling is the ability to strip away the unnecessary details to truly understand a situation. For our purposes, arbitration works like this. Baseball arbitration is an adversarial game between two parties which will now be called ‘ownership’ and ‘the player’. We assume that the baseball player has some underlying true value in dollars. Ownership and the player each submit a salary to the arbiter and he selects the value which is closest to the player’s true value. The goal of ownership is to minimize the amount the pay to the player while the player’s goal is to maximize his own value.

If a player sells himself short, he may lose out on money the arbiter thinks he deserves. If the player values himself too highly, the arbiter will be more likely to rule in favor of the ownership, costing the player potentially millions. What strategy should the ownership and the player use to determine the value they submit to the arbiter?

Baseball Arbitration Game Theory: Simplest Case

As I mentioned above, mathematicians solve problems by first studying the simplest possible incarnation of a problem and then increasing the difficulty. First, we need to understand the basic mechanism of action. Then, we can get rid of unnecessary simplifying assumptions. We iterate solving and increasing the difficulty until we come up with a model that well-approximates reality. Here, we start with the simplest case where the ownership, the player, and the arbiter all know the player’s true value in dollars.

Some Concepts in Game Theory

Game theory is the branch of mathematics that studies decision making in adversarial games. Game theory has obvious applications in traditional games like poker. However, one of the main giants in the field of game theory – the great John Nash of A Beautiful Mind fame – first developed his techniques in order to use game theory to study economics. And why not? Economics is certainly adversarial at its heart. Businesses want to set their prices to maximize profits (unfortunately) at the expense of their customers. The wealthy want to buy and sell socks at the perfect times so as to maximize their gains at the expense of other stockholders. The language and knowledge of game theory has far reaching applications including our problem concerning baseball arbitration.

One of the key ideas in game theory is that of a Nash Equilibrium. A Nash equilibrium is a choice of strategy for player A and for player B. The two strategies are said to be in a Nash equilibrium if either player deviating from their strategy actually results in a worse outcome. For example, A and B’s strategies are in Nash Equilibrium if, while player B’s strategy remains constant, player A can’t change his strategy to benefit himself (and vice versa!). If both players use strategies that are in a Nash equilibrium, then there is no incentive to either player to change their strategy. Hence, the nomenclature equilibrium: strategies at the equilibrium tend to remain there.

Identifying Nash equilibria is often a good way to analyze games to determine optimal methods of playing. In the next section we will identify the Nash equilibrium of our baseball arbitration game.

Nash Equilibrium for Simple Baseball Arbitration

In the simplest case – when both parties and the arbiter know the player’s true value to be $x – the optimal strategy is very boring. Indeed, both the ownership and the player should submit precisely $x to the arbiter. The way to determine if a pair of strategies are in a Nash Equilibrium is to ask ‘could either player benefit by changing their strategy?’ If we can argue that neither the ownership nor the player can improve their expected gains by changing their strategy, we have shown that both parties submitting the true value is a Nash equilibrium.

Remember that the arbiter will choose the contract value that is closest to $x. If the ownership submits $x as a bid, the arbiter will award a contract valued at $x. Therefore, there is no benefit to the player to submit any value other than $x. The same logic applies to show that there is no value gained if the ownership submits a value other than $x. Therefore, the Nash equilibrium exists when both parties submit the player’s true value to the arbiter.

Suppose, on the other hand, that the ownership submits something else. Suppose the baseball player’s value is $4 million but the ownership submits $3 million. Then, the player can ensure they are paid over their true value by submitting anything between $4,000,001 and $4,999,999 to the arbiter. Therefore, truly the optimal solution here – the setting with perfect knowledge – is for everyone to submit the true value.

One can notice that while there is no value gained by, say, the ownership submitting less than $x when the player submits precisely $x, the ownership also doesn’t lose any money by bidding less than $x. This scenario is called a weak Nash equilibrium.

The fact that our equilibrium is weak combined with the improbability of all three parties knowing precisely the value $x invites us to explore more complex and realistic scenarios in the coming discussion.

The Case of the Unpredictable Arbiter

As I championed above, let us now add just one slight wrinkle of difficulty in our analysis to work up to a realistic situation. Here we assume that both the ownership and the player know the player’s true value to be $x. However, the arbiter here is unpredictable and may not determine that player’s value to be $x. To model the uncertainty in the arbiter’s valuation, we assume that the arbiter’s determined value is a random variable that is symmetric and whose mean is the player’s true value.

What is the optimal strategy here?

It turns out that this problem has actually been studied by other mathematicians and it is known that without assuming more about the probability distribution, there is no optimal strategy. One of my favorite parts about mathematics is that the difference between a trivially easy problem – like the case where we know the arbiter’s valuation – and an impossibly difficult one is almost invisible. I am not the only person who has observed and enjoys this phenomenon.

So what can we do? Well, clearly we added too much difficulty into the problem at once. We need to take a step back into an ever so slightly easier setting to see if we can say anything meaningful. We need to assume that the arbiter’s valuation can be represented with a more concrete distribution. The following few sections will be heavily mathematical (almost as bad as our prior article about fouling in basketball) but skipping ahead to the ‘Role of Knowledge in Baseball Arbitration’ section will summarize my findings without the burdensome mathematics. For those mathematically inclined, though, we trudge on.

Some Analysis of the Prior Section

Suppose that the submitted bids to the arbiter by the ownership and the player are z and x, respectively, while the true value of the player is \mu . We assume here (accurately!) that the ownership bids less than the player – in math that z<x. Then, because the arbiter’s action is randomized, we will compute the average contract value in terms of x and z. In the following analysis, we let the function \varphi (t) represent the probability density function for the Gaussian N(\mu, \sigma^2). Then, the expected contract value, denoted by \mathbb{E} is \mathbb{E}=z\int_{\infty}^{\frac{x+z}{2}}\varphi(t)dt+x\int_{\frac{x+z}{2}}^\infty \varphi(t)dt.

This doesn’t tell us much, but let me show you what \mathbb{E} can look like for fixed ownership bid z and various x values. We will return to the mathematics shortly, but for the experiments we need to give some proper values to our variables. Suppose that our player has a true contract value of \mu = $4.2 million but the ownership bids z = $3.3 million. Moreover, we assume can estimate that there is a 95% chance the arbiter returns a valuation between $3.4 million and $5 million. It turns out that if the player bids the correct amount (rather, in the correct range), their expected contract value will be over $4.2 million.

The following graphic shows the expected contract value for a variety of player bids in this setting. The green dots correspond to average contract values that are larger than the player’s true value. The red dots are when the player receives less than their true value on average. Players want to submit bids in the green range.

Notice that in this situation, when the ownership severely undervalues the player (we will actually define severely later!), the player is rewarded when the deviation between the true value and his bid is less than the deviation between his true value and the ownership’s bid.

The cutoff from green back down to red on the right side happens at a player bid of $5.1 million. Notice this $5.1 million corresponds precisely to the player overbidding by the same amount as the ownership underbid (a difference of $0.9 million). In fact, the player, in this setting, should place a more conservative bid. Bidding somewhere above his true value, but not as far away as the ownership’s bid will lead to the player receiving more money on average.

What if the ownership is fairer and underbids by a much smaller amount? Suppose that again the player’s true value is $4.2 million but now the ownership bids $4 million. Here, the ownership is submitting a pretty reasonable bid.

Notice here that the range of values that correspond to our player gaining value over his true value (the green dots) goes from $4.4 million to about $5.1 million. In this setting, the ownership underbid but not severely so. In this case, the player is rewarded by being greedy, by overbidding by more than the ownership underbid.

Conclusion/Observation number one: If the ownership is greedy and severely underbids, the player should be relatively conservative and submit a bid to the arbiter that is close to his true value. If the ownership is conservative and bids close to market value, then the player should be greedy and overbid by more than the underbid margin on the ownership’s side. When the ownership’s strategy zigs, the player should zag.

As a mathematician, I have been trained to ask and answer the natural question: What is ‘severe overbidding’?

Severe Underbidding and the Transitional Bid

In the prior section we observed that two things can happen in this game. Two distinctly different phenomena occur based on whether the ownership severely underbids the player’s value or not. However, the term ‘severely overbid’ was not precise. As a mathematician, I am naturally led to investigate what constitutes a severe underbid. I will make the following observations:

• No matter the ownership’s bid, z, the player can always submit a particular bid, x=2\mu-z which I will call the ‘mirror bid’ which
  • Corresponds to the player overvaluing himself by the same amount the ownership undervalued him, and
  • Leads to an expected contract value of precisely \mu by a nice symmetry argument!
• ‘Sever Underbidding’ for the ownership can be recognized by observing that the slope at the mirror bid is negative: the player should be more conservative. If the ownership does not severely underbid, the slope of the curve at the mirror bid will be positive.
• At the transition between severely underbidding and not, the slope of the curve at the mirror bid should be precisely 0. This indicates that the mirror bid is the optimal response for the player. In fact, this pair of bids – the ownership and the player submitting bids at the transition between severe over/under bidding and not – will be our Nash Equilibrium!

Since my method of analysis is slope-based, we need to use partial derivatives to solve for the transitional bid.

Recall our notation: \mathbb{E} is the expected salary value, z is the ownership’s bid, x is the player’s bid, and \varphi is the normal distribution. Differentiating our previous formula for expected value (and using the fundamental theorem of calculus!), we get: \frac{\partial \mathbb{E}}{\partial x}=(\frac{z-x}{2})\varphi(\frac{x+z}{2})+\int_{\frac{x+z}{2}}^\infty \varphi(t)dt.

Solving for when this partial derivative equals zero indicates that the transitional bid for the owners is z=\mu -\frac{\sigma\sqrt{2\pi}}{2}. If the ownership submits this bid, the maximum contract value the player can achieve is precisely \mu, his true value. If the ownership submits anything other than this transitional value, the player can submit a bid that results in an expected profit.

That means the ownership is discouraged from submitting any bid other than this transitional value. Below, I’ll include a few experiments verifying our results.

In the following, the player has a true value of 4.2 million dollars and the arbiter’s uncertainy can be measured by assuming there is a 95% chance he returns a bid between 3.4 and 5 million. Then, the ownership’s optimal bid (the transitional bid) is 3.6884 million. The player’s expected contract value curve is shown below.

Notice that no matter the player’s bid, if the ownership submits this transitional value, the player cannot earn ‘above his true value’.

Last, let us reverse the direction of analysis. What if the player submits their own bid at the ‘transitional value’ between severe overbidding and conservative overbidding. Below is his expected value curve where the x-axis is now the ownership’s bid. The transitional bid for the player is 4.712 million in the same setting (which, notice, is the location of the peak in the previous graph).

Notice: if the player submits the transitional bid, he is guaranteed a contract value above his true value. Here, the best the ownership can do is submit 3.6884 million (their transitional bid!) to force the contract value to be exactly 4.2 million.

The symmetry is beautiful.

Some Final Thoughts

The most immediate observation I have is that in the case where \sigma = 0 our analysis reduces to the ‘perfect arbiter’ case above. If the uncertainty is 0, then we know precisely how the arbiter will rule. Here the optimal strategy is to bid precisely your true value. This was the conclusion we reached at the beginning of this article.

Nash Equilibrium are only optimal when both players operate within the equilibrium. If the player or the ownership deviate from this optimal bid – the transitional values – then the other party can seek to profit from this inaccuracy. If, for instance, the ownership deviates from the equilibrium strategy, the player can profit. In order to know their new optimal bid, though, the player needs to know precisely the ownership’s bid. This is unlikely.

However, if the ownership deviates from the optimal strategy and the player stays with the optimal strategy, they will still benefit. The player will, in this case, be rewarded with a contract that is larger than their true value. Truly, this is a stable equilibrium because even if one side deviates, it is in the other side’s best interest to stay the same (unless they have perfect information, which they do not).

One final thought: while this analysis essentially solves baseball arbitration, there is one huge caveat which may make this analysis not-that-useful. We made two assumptions about the arbiter. First, that on average they will return the player’s true value. This is not that unreasonable, though one may debate what ‘true value’ is. The second is this: in order to determine the optimal bid we need to know the uncertainty in the arbiter’s valuation. Perhaps one could do this by studying past arbitration cases. But, because arbitration is seen as so undesirable, there are not tons of examples. In order to use this analysis, one needs to make a guess at the uncertainty. Unfortunately for individual players, this is something that ownership is much more likely to be good at.