IS MANDINGO FIGHTING AN ECONOMICALLY RATIONAL ACT? 

Problem

 Edna Greene Medford, has been quoted in a number of articles (http://www.nextmovie.com/blog/is-mandingo-fighting-a-real-thing/) concerning the film Django Unchained for her conclusion that economic rationality would have prevented the practice of ‘mandingo fighting’ in pre-civil war America given that : “enslaved people are property, and people don't want to lose their property unless they're being reimbursed for it," 

This conclusion, however, relies on a presupposition that is counter to traditional models of economically rational actors (that they do no risk their property unless they are being reimbursed). Indeed, this is a characteristic of an economically irrational actor (who will only make investments if they are certain to result in a 100% return)

Rejecting her assumption, however, her conclusion might still be tested with the following question ; Is mandingo fighting a rational economic action.

Summary of Results

If third party ‘insurance wagers’(wagers made between the slave owner and a third party) are not allowed, then mandingo fighting is an economically rational act if :

(-y) + ((-y)(-x₁)) + 80,000x – x₂y ≥ 40,000

Where:  y is the amount wagered on the fight

x₁  is percentage belief that one slave owner has that their slave will win the fight

x₂  is percentage belief that the other slave owner has that their slave will win the fight

x₁ does not equal (1-x_2), and

percentage belief that the other slave owner has that their slave will win the fight

            40,000 is accepted as the average value of a slave in todays dollars (http://thecnnfreedomproject.blogs.cnn.com/2011/04/06/bales-average-price-of-slave-has-decreased/)

If third party ‘insurance wagers’ are permitted in mandingo fighting, then, given the appropriate wager, it is always an economically rational action, except in the case where a slave owner has a 0% belief in their slave’s ability to win the fight.

Assumptions

We will start with the presumption that slave-owners are economically rational actors. This is a useful assumption for our present inquiry for two reasons. First, although theories of predictive bounded irrationality can provide some useful projections of how individuals will act in a modern economy, they begin to lose their integrity when nearly every one of their fundamental assumptions requires tweaking to address pre-Civil war society. Second, if we reject the premise that slave owners were economically rational actors then, the point is conceded that they might engage in mandingo fighting simply for the irrational malice and the argument that mandingo fighting would have been prevented out of economic self-interest is indicted.

We will continue with a second presumption concerning economically rational actors. That un-invested money and the potential for monetary grain through investment are treated as equally weighty in the mind of an economically rational actor. Put another way, an economically rational actor is not impact by ownership bias, or status quo bias. For example, an economically rational actor will when presented with an investment opportunity with a 51% chance of 200% return on investment and a 49% chance of 0% return on investment will weight that investment as worth more than the investment money and take the investment opportunity as often as possible. This in contrast to a non-rational economic actor that is risk-averse and weight the risk of a 0% return on investment too heavily and not make this rational investment.

However, if an investment opportunity offered only a 50%  chance of a 200% return on investment, the economically rational investor would be neutral on the investment.   This is in contrast to a risk-loving and irrational economic actor that may way the risk of a 50% chance of a return on investment too heavily and make this irrational investment.

Consider the decision expressed mathematically:

$100 of un-invested money (ignoring depreciation) is worth $100. Therefore, a rational economic actor will take any investment opportunity that is worth more than $100.00

$100 of investment money in the first investment scenario is worth:   (.51 x $200) + (.49 x $0) = $102. This investment opportunity is work more than $100.00 and investment is rational.

$100 of invested money in the second investment scenario is  worth : (.50 x $200) + (.50 x $0) = $100. This investment opportunity is not worth more than $100 and is, therefore irrational.

Analysis

According to Kevil Bales, the average price of a slave of 1809 was $40,000 (in todays’ dollars).  In the above model. This can be considered ‘un-invested money.’ It is invested in the body of the slave and, thus, worth the value of their labor, however that labor should not be included in the calculation of the value of the salve in a mandigo fight because it can be re-purchased for (on average) $40,000.

Therefore a economically rational slave-owner when presented with a mandigo fight investment opportunity will should take the opportunity in certain circumstances (when the opportunity is worth more than the value of the invested money : $40,000).

Whether the investment is worth more than $40,000 depends upon a reasonable prediction of the outcome of the fight. Because, as noted in  a number of reviews, these fights are “battles to the death”, they present a fairly simply game theory model (because there are only two possible outcomes : dead slave (100% lost of investment), alive slave (no loss of investment).

The potential outcomes are illustrated below :

	Slave Owner 1 – Dead Slave	Slave Owner 2 -  Live Slave
Slave Owner 2 – Live Slave	Outcome 1 Slave Owner 2 has at least 100% return on investment (the slave investment + any additional wager)	Outcome 2 Impossible in “battle to the death”
Slave Owner 1 – Dead Slave	Outcome 3 Impossible in a “battle to the death” ? (I suppose the slaves could both kill each-other)	Outcome 4 Slave Owner 1 has at least 100% return on investment (the slave + any additional wager)

If either slave-owner has wagered at least one dollar, and there is a 51% chance that their preferred outcome will result, then they should take the opportunity to invest. What if, however, they have wagered more than one dollar? Consider for example they have wagered $120,000 (the value of 3 slaves?).

That presents the following equation in the mind of the economically rational salve owner (where x is the percentage of their slave winning the mandingo fight) :  

When is x($120,000) + (1-x)(-40,000) > 40000.

The solution, as you probably guessed, is anytime the chance that x > 1/3, it is economically rational to engage in the fight.  Meaning that given the appropriate wager, mandingo fighting with even a slave that is very likely to lose is a rational investment (continuing the logic, if your slave is a 5 to 1 underdog you will need to make a $200,000 wager, if your slave is a 6 to 1 underdog you will need to make a 240,000 wager).

It is unclear from descriptions of mandigo fighting whether the betting must be only between the slave owners or whether side investments between slave owners and third parties are permitted (assuming a third party wager is permitted then every mandingo fight can become an economically rational choice).

However, if the wager must be between slave owners (with slaves in the fight), then the question becomes : when will a deal be formed between the two. Contract theory provides us a model for assessing whether two parties will come to a deal, and it essentially boils down to whether there is a incongruity of beliefs between the parties, whose margin of error is within the range that renders a potentially positive outcome for both parties.

Explaining this in the above terms, another economically rational slave-owner would not take the $120,000 bet because his percentages are reversed if he believes the outcome of the match is the same as his opponent. His calculation looks like this (where x is the agreed likelihood of winning the match)

(1-x)(-120,000) + x (40,000) > 0

This is precisely the opposite of the other slave-owner so no deal will be made here.

However, if there is a differential in the belief of the outcome then two economically rational actors may make the deal. Consider the difference in beliefs that is necessary for a $120,000 wager to be rational for other sides.

One side must assess that (1-x)(-120,000) + x (40,000) > 0

This reduces to x > 2/3

The other must assess that (x)(120,000) + (1-x)(-40,000) > 0

This reduces to x > 1/4

 Therefore, in order for this mandingo fight to happen between economically rational actors, the first slave owners must believe his or her slave is at least a 67% (rounding up) chance to win, and the other must believe they are at least a 25% chance to win. There is clearly a gulf of information here 67 + 25 = 93, there is a seven percent overlap in their predictions, on when such an overlap occurs, a deal may rationally be struck between two parties with divergent information.

This can be abstracted to take into account the all variables, when x₁ is the percentage belief that the one slave-owner has in the outcome that their slave will kill the opposing slave, x₂ is the percentage belief of the opposite result and y is the proposed wager :

(-y) + ((-y)(-x₁)) + 80,000x – x₂y ≥ 40,000

Conclusion

Given the appropriate circumstances, mandingo fighting is an economically rational act.