Reasoning it Through...

I just finished reading Two Person Game Theory by Anatol Rapoport. I'll freely confess that much of the mathematics was over my head, despite his admirable restraint in limiting his discussion as closely as possible to basic algebraic equations. But, in an effort to test my absorption, I'd like to tentatively apply the basic tenets of game theory to the practice of plea bargaining to see if we can rationally predict the outcome of policy decisions in the criminal justice process.

So, let's take the case of Patsy Jarrett, who claims to be entirely innocent of charges for murder and robbery. Since the court system is adversarial in nature, the other "player" in this scenario would be the Prosecutor. Since the payoffs don't have a strict correlation (the prosecutor's interest in winning a conviction is not exactly identical to the defendant's interest in avoiding a life sentence), it would be a non-zero sum nonnegotiable game. Now, one of the cool tools of game theory is the game matrix - in this game, our matrix would look like this:

	D1	D2
P1	(a, w)	(b, x)
P2	(c, y)	(d, z)

P₁ and P₂ would represent the two strategies available to the Prosecutor - Negotiate or Prosecute. D₁ and D₂ would represent the similar strategies available to the defendant - Negotiate or Prosecute. The utility values at each location on the game matrice would represent the value to each player (a-d for prosecutor, w-z for defendant) of the outcome attaining at that overlap of the two strategies.

However, there's another factor which we should also consider, so I'll just throw it in here now - which is the probability of one side prevailing over the other if the case should actually go to trial - i.e., there is some probability that the Prosecutor would win, and some probability that the defendant would win in a trial outcome. Since P's taken, let's use the value N to signify the probability that the prosecutor will prevail in a trial. Also, since the values of winning and losing are the same in all instances where the case goes to trial, we can also simplify the chart considerably by using similar values in each case:

	D1	D2
P1	1(a, x)	N(b, y); (1-N)(c, z)
P2	N(b, y); (1-N)(c, z)	N(b, y); (1-N)(c, z)

OK. So that gives us a matrix... now, let's see if we can come up with some applicative values for the corresponding "utilities" in the chart. Let's stipulate that the prosecutor fully believes in the guilt of the defendant, thus eliminating any utility cost to the prospect of placing the wrong person in prison. From the prosecutor's perspective then, his interests are securing a conviction. To a large extent, his personal motive (say career advancement) would also be entirely congruent with a justice motive (since he believes justice will be served by a conviction). So, we could estimate that his value for securing a conviction is 1 and that his value for failing to do so is -1. If that were the case, then a=b, since the prosecutor is indifferent to whether the defendant is convicted or pleads. However, let's assume that the prosecutor also has an interest in securing the longest possible prison sentence. Thus, his potential outcomes are 25-life in the case of a conviction, 10-15 years for a plea bargain, and 0 years for an exoneration in open court. We can then say with confidence that, of the potential prosecutorial outcomes, the preference for b > a > c. Since c is also defined in this context as a negative outcome, we also know that c < 0.

The outcomes of the defendant are identical to those of the prosecutor (25-life, 10-15, and 0). Intriguingly, however, receiving a sentence of zero years is hard to really classify as a payoff. Though there may be some positive value to securing an exoneration, surely this would be largely offset by the costs of the trial and trauma of being falsely accused. It may be better to be falsely accused and exonerated than to be falsely accused and convicted, but neither is preferable to not being accused at all. Thus, in the defendant's case, 0 > z > x > y.

So one thing that starts to loom large in this picture is the importance of that value N.

Intriguingly, this value appears to be highly dependent upon the judge presiding over the trial. The article linked cites the various conviction rates of judges dealing with one specific class of cases (DWI) within a single county (Mecklenberg County, North Carolina). They range from a high of 86% to a low of 40%. Though it may be an unjustified assumption, let's assume there are similar discrepancies throughout the system.

Now, I want to arbitrarily assign numeric values to the preferences (because I suck at math) so I'm going to propose a practical method for assessing the defendant's preference values - let's say that each of her negative preferences is conditioned by the amount of life the outcome will "cost" her. Thus, if we estimate that the average length of a trial is one year and the defendant's estimated remaining lifespan at the time of trial is 50 years, the most preferable outcome (exoneration) will have a value of -0.02. A plea bargain will have a cost ranging from -0.2 to -0.3. And a conviction will have a cost ranging from -0.5 to -1. If its rational for the decisionmaker to assume the worst outcome under each probability, we then get values of x=-0.3, y=-1, z=-0.02

Now, the way this game is set up, the moves are not made simultaneously, but rather in sequence. The defendant cannot unilaterally plea bargain - the prosecutor must offer a plea. So, the defendant has perfect knowledge of the prosecutor's first move. The only time the defendant can really make a choice is when the the Prosecutor selects strategy P₁. At this point, the Defendant is forced to choose between D₁ and D₂. Now, let's say that the defendant is facing a judge with a conviction rate of 50%. We now have the following decision matrix:

	D1	D2
P1	-0.3	(0.5) * (-1); (0.5) * (-0.02)

So, on average an innocent defendant would receive a "payout" of -0.51 for choosing the trial and a guaranteed payout of -0.3 for choosing the plea bargain. Thus, it would be irrational to choose the trial. In order to provide a rational incentive for the innocent defendant to brave a trial, the conviction rate (ideally, of innocent defendants, realistically that figure isn't available) would have to be low enough that the average of all defendants would do better on a trial than a plea bargain. If I'm not mistaken (seriously big if), the conviction rate would have to be no more than 27%. Given that the conviction rate is far higher than this, and assuming my assumptions of relative costs aren't too far out of whack, this would help explain why 95% of all defendants decide to plead... even if you're innocent, the most rational decision is to take the plea bargain.

I'd like to develop further the policy ramifications of this... but I think I need to flesh it out more fully (taking a stab at the prosecutorial incentive structure, double-checking my math, etc.).

More later...