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Game Theory and the HumanitiesBridging Two Worlds$

Steven J. Brams

Print publication date: 2011

Print ISBN-13: 9780262015226

Published to MIT Press Scholarship Online: August 2013

DOI: 10.7551/mitpress/9780262015226.001.0001

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Law: Supreme Court Challenges and Jury Selection

Law: Supreme Court Challenges and Jury Selection

(p.127) 6 Law: Supreme Court Challenges and Jury Selection
Game Theory and the Humanities

Steven J. Brams

The MIT Press

Abstract and Keywords

This chapter begins by analyzing two games that were played between the president and the Supreme Court. More specifically, it describes a game played between President Richard M. Nixon and two of his appointees to the Supreme Court, who clashed over the release of White House tape recordings made during the Watergate crisis of 1974. The chapter shows that by threatening not to release the tapes describing an attempted cover-up of a burglary that he had instigated, Nixon precipitated his own downfall. It then turns to how President Franklin D. Roosevelt attempted to “pack” the Supreme Court by appointing additional justices who would favor New Deal legislation, but his strategy backfired and he was forced to retreat. The chapter also develops a model of jury selection by the prosecution and defense, based on a two-person constant-sum game.

Keywords:   game theory, president, Supreme Court, Richard Nixon, Watergate, Franklin D. Roosevelt, New Deal, jury selection

6.1 Introduction

The three traditional branches of government in the United States— the Presidency, the Congress, and the Supreme Court— have never been entirely at ease with each other. The most persistent and rancorous conflicts have been between the president and Congress, especially when one or both of its two houses (the House of Representatives and the Senate) are controlled by a different party from that of the president.

The U.S. Constitution allows the president to veto bills passed by Congress, but Congress can override a presidential veto by two-thirds majority votes in both its houses. Consequences of these rules for the distribution of power among the president, the House, and the Senate are analyzed in Brams, Affuso, and Kilgour 1989. In this federal system, the Supreme Court is also a significant player, because it can, by a simple-majority vote, declare laws passed by Congress and signed by the president unconstitutional.

In this chapter, I begin by analyzing two games that were played between the president and the Supreme Court, whose members the president nominates and whose nominations must be ratified by the Senate. More specifically, in section 6.2, I describe a game played between President Richard M. Nixon and two of his appointees to the Supreme Court, who clashed over the release of White House tape recordings made during the Watergate crisis of 1974. In section 6.3, I show that by threatening not to release the tapes describing an attempted cover-up of a burglary that he had instigated, Nixon precipitated his own downfall.

(p.128) Nixon’s threat was not the first time a president had challenged the Supreme Court’s authority to interpret the laws and determine their constitutionality. However, his threat is the only one ever to lead to the resignation of a president. Earlier, President Franklin D. Roosevelt had vehemently disagreed with some of the rulings of the Supreme Court on the constitutionality of his New Deal legislation. In section 6.4, I describe how he attempted to “pack” the Court by appointing additional justices who would favor New Deal legislation, but his strategy backfired and he was forced to retreat.

Although both Nixon’s and Roosevelt’s challenges ultimately failed, the games these presidents played with the Supreme Court are quite different. In fact, I use different game theory models— noncooperative in the case of the Nixon, cooperative in the case of the Roosevelt— to explain these failures.

In both cases, I focus not on the Court as a single player but on specific members of the Court and the choices they made. In the case of the Roosevelt Court, I illustrate how the voting power of different ideological blocs on the Court can be measured.

In section 6.5, I develop a model in which the prosecution and defense, before the commencement of a trial, attempt to select a jury that they believe will be as favorable to their side as possible, based on a two-person constant-sum game (so that what is best for the prosecution is worst for the defense, and vice versa). Because the calculations are complex, however, I give only examples illustrating a player’s optimal choices.

The analysis provides insight into the kinds of strategic calculations attorneys make in selecting a jury, which I illustrate with a well-known case in the 1970s. But the analysis also pinpoints a procedure that would make such calculations irrelevant and, I argue, would lead to more impartial juries. In section 6.6, I summarize the results and offer some thoughts on treating law as a subject in the humanities.

6.2 The White House Tapes Case

When Richard Nixon resigned his presidency on August 9, 1974, the most immediate cause was a decision in United States v. Nixon, handed down on July 24 by the Supreme Court, ordering him to release certain (p.129) White House tapes.1 These were recordings of conversations relating to a break-in at the Watergate housing complex in Washington, DC, which had been planned in the White House, and the subsequent attempt to cover up this crime after it was discovered.

The Court unanimously ruled that Nixon must cease his efforts to withhold the tapes from Special Prosecutor Leon Jaworski in the so-called Watergate cover-up case. That same day the president, through his attorney, James St. Clair, announced his compliance with the ruling— he would release the White House tapes that the special prosecutor had sought.

Fifteen days later, the Nixon presidency ended in ruins, a direct result of the Court’s action. This case is of special interest, because optimal strategies in the game played over the release of the tapes, as I reconstruct it, led to a paradoxical consequence— an outcome worse for both sides than a more “cooperative” outcome in the game— that seems to challenge the rationality of the players’ choices.

The history of the White House tapes decision began on March 1, 1974, when a grand jury indicted seven former White House and campaign aides for attempting to cover up the Watergate burglary (United States v. Mitchell et al.). On April 16, the special prosecutor petitioned Judge John Sirica in the federal district court of the District of Columbia to subpoena tapes and documents of sixty-four presidential conversations Nixon had with advisors John Dean, Robert Haldeman, John Erlichman, and Charles Colson; Judge Sirica issued this subpoena on April 18.

The prosecutors moved quickly to prevent delay. On the day that an appeal by St. Clair was filed in the Court of Appeals, Leon Jaworski, using a seldom invoked procedure, went to the Supreme Court and sought a writ of certiorari before judgment that would leapfrog the appeals process. Citing the imminent cover-up trial date, Jaworski also noted the necessity to settle expeditiously an issue that was paralyzing the government. He requested the Court not only to issue the writ but also, because of the “imperative public importance” of the case, to stay in session into the summer (Lukas 1976, 495).

(p.130) That way the case could be decided in sufficient time for the tapes to be used as evidence at the cover-up trial, should Judge Sirica’s ruling be upheld. The Supreme Court agreed on May 31 and heard oral arguments on July 8.

When the Supreme Court justices went into conference on July 9, each of the eight justices who were to consider the case had basically two choices— decide for or decide against the president.2 It appears from the record that six of the justices reached an early consensus against the president on all three of the major issues: (1) whether the Court had jurisdiction in the case— legal standing to sue— since Jaworski was an employee of the executive branch; (2) whether executive privilege was absolute; and (3) whether Jaworski had demonstrated a sufficient need for the subpoenaed materials.

Justices Warren E. Burger and Harry A. Blackmun, while concurring with the majority on limiting executive privilege, believed that the special prosecutor lacked standing to sue the president. For this reason, it appears, they voted originally against granting the case certiorari (Totenberg 1975).

Justices Burger and Blackmun are conceived of as one player. This is because it was almost axiomatic that Blackmun voted with Burger: In the first five terms (1970–1974) that Burger and Blackmun served together on the Court, they agreed on 602 of the 721 cases they both heard (83.5 percent), which was the highest agreement level of any pair of justices who served over these five terms.3 They were referred to as the “Minnesota Twins” by the Supreme Court staff.

As deliberations proceeded, Burger and Blackmun had a choice of two strategies:

  1. 1. To decide for the president (F), forming a minority to create a 6–2 “weak” decision; or (p.131)

  2. 2. To decide against the president (A), joining the other six justices to create a unanimous decision.

President Nixon’s possible response to an adverse Supreme Court ruling was long a matter of doubt— and one Burger and Blackmun could not afford to ignore. On July 16, 1973, White House Deputy Press Secretary Gerald Warren stated that President Nixon would abide by a “definitive decision of the highest court.” Nixon, at a news conference on August 22, 1973, endorsed the Warren formulation, but neither he nor White House spokesmen would expand on the original statement (New York Times, July 22, 1974, 18; Time, July 22, 1974, 15–17). These statements were made in reference to the president’s refusal to obey a subpoena from the first special prosecutor,Archibald Cox, for nine White House tapes.

That case never reached the Supreme Court. The Court of Appeals ruled against the president, who, after a delay of eleven days, agreed to submit the tapes, but not before he had dismissed Cox. The question of what “definitive” meant then became moot.

The issue arose again on May 24, 1974, when Cox’s successor, Jaworski, filed his appeal with the Supreme Court. On July 9 St. Clair made it clear that the president was keeping open the “option” of defying the Court. The question of compliance, St. Clair stated, “has not yet been decided” (New York Times, July 10, 1974, 1).

Since the expectation at the time was that the Court would rule against the president (Newsweek, July 22, 1974, 18; Time, July 22, 1974, 15–17), President Nixon had two strategies:

  1. 1. Comply (C) with an adverse Court ruling; or

  2. 2. Defy (D) an adverse Court ruling.

Several factors help to explain President Nixon’s refusal to make an unambiguous commitment concerning his response to a Court decision. If he stated that he would not comply, his statement might be used as a ground for impeachment. If he stated that he would comply, the House Judiciary Committee might argue that the president would either have to comply with its subpoenas, too, or be impeached (New York Times, July 10, 1974, 1).

(p.132) More important, though, the president’s refusal to assure his compliance with an adverse decision was designed to threaten the Court and lead the justices to render either a favorable decision or, at worst, an adverse, but closely divided, split decision that Nixon could claim was insufficiently “definitive” for a matter of this magnitude. Evans and Novak (1974, A29) noted at the time, “The refusal of St. Clair to say Nixon would obey an adverse decision has disturbed the judicial branch from the high court on down.”

If the president’s intent was to threaten the Court, the threat backfired. Why? To explain why Justices Burger and Blackmun departed from their apparent personal preferences and eventually sided with the Court majority, I next describe the game they and Nixon played.

The possible strategies of the two players (F and A for Burger and Blackmun; C and D for Nixon), and the probable outcomes and their rankings by the players, are shown in figure 6.1, starting from the upper left-hand cell and proceeding clockwise.

  1. I. FC: Burger and Blackmun pleased, Nixon satisfied— (4,3) A constitutional crisis is averted and the majority-rule principle (to be described) is preserved; Nixon is not impeached for noncompliance.

    Law: Supreme Court Challenges and Jury Selection

    Figure 6.1 Payoff matrix of White House Tapes Game (game 27)


  2. II. FD: Burger and Blackmun dismayed, Nixon happy— (1,4) A constitutional crisis ensues because of Nixon’s defiance; Nixon is impeached by the House, but his conviction by the Senate is uncertain.

  3. III. AD: Burger and Blackmun unhappy, Nixon defeated— (2,1) There is a constitutional crisis, but it is resolved by the impeachment and certain conviction of Nixon.

  4. IV. AC: Burger and Blackmun satisfied, Nixon unhappy— (3,2) A constitutional crisis is averted, but the majority-rule principle is weakened; Nixon is not impeached for noncompliance but is forced to resign.

To justify these outcomes and their rankings, first consider the consequences associated with Nixon’s defiance of an adverse Supreme Court ruling (AD). Unquestionably, if the president defied the Court, his defiance would represent a direct assault on the Supreme Court’s constitutional place as the “principal source and final authority of constitutional interpretation” (Stephenson 1975, 292) and thereby threaten the very structure of the American political system.

Indeed, it seems highly probable that Nixon would have plunged the country into its deepest constitutional crisis since the Civil War. No previous president had ever explicitly defied an order of the Supreme Court, though such action had apparently been contemplated (Scigliano 1971, chap. 2).

At the time of the decision in United States v. Nixon, it appeared that the result of presidential defiance would be impeachment by the House on the ground of withholding evidence from the special prosecutor or violation of the principle of separation of powers. While the outcome of a trial in the Senate was less certain, a unanimous adverse decision (A) by the Court that included three conservative Nixon appointees (Burger, Blackmun, and Lewis F. Powell, Jr.) would preempt changes that the president was the victim of what presidential counselor Dean Burch called a “partisan lynch mob” (Lukas 1976, 510).4 Pointedly, on the day of the decision, St. Clair warned the president that he would surely be (p.134) impeached and swiftly convicted if he were to defy the unanimous ruling of the Court (Ibid., 519).

Yet, Jaworski believed, “if the vote against [the president] was close he would go on television and tell the people that the presidency should not be impaired by a divided Court” (Jaworski 1976, 164). A “weak” decision from which at least some of the more conservative Nixon appointees dissented would also allow the president to continue his “one-third plus one” strategy in the Senate (FD) to avoid conviction and removal from office (by a two-thirds or greater majority).

Consider next the consequences associated with Nixon’s compliance with an adverse Supreme Court decision (AC). Clearly, compliance would avert a constitutional crisis, and Nixon would thereby avoid immediate impeachment in the House for not complying with the Court. However, compliance posed problems for the president: He had reason to believe that the subpoenaed material, if released, would prove damaging and might even lead to his eventual impeachment by the House. In fact, upon hearing of the Court’s decision, Nixon, who was at his San Clemente, California, home, telephoned White House Special Counsel Fred Buzhardt in Washington, saying, “There may be some problems with the June 23 tape” (Washington Post, September 9,1974, A1).

Although the revelation of this tape ultimately forced his resignation, Nixon apparently did not fully realize at the time the incriminating nature of the recorded conversations. Woodward and Bernstein (1976, 176), on the one hand, report that Buzhardt felt that the tape was “devastating.” Nixon, on the other hand, felt that Buzhardt was “overreacting,” that it was “not that bad.” Even as late as August 5, in his statement accompanying the public release of the tape transcripts, Nixon reflected his evaluation of the tape’s impact:“I recognize that this additional material I am now furnishing may further damage my case… I am firmly convinced that the record, in its entirety, does not justify the extreme step of impeachment and removal from office” (New York Times staff 1974, 324).

Compliance— or, more accurately, the announcement of compliance— would allow the president to fall back on his long-used strategy of delay, though it would not necessarily remove the threat of impeachment and ultimate conviction, especially if the Court was unanimous in its judgment. For Justices Burger and Blackmun, who had voted originally (p.135) against granting the case review, supporting and enlarging the majority (A), possibly against their convictions, to counter a presumed threat to the Court’s authority might weaken the majority-rule principle that any majority is sufficient for a decision to have the force of law. But voting their convictions (F) would be hazardous should the president use a divided decision as a pretext to defy the Court.5

These conflicting considerations can be combined to yield the ranking of outcomes that I posited earlier. I start with President Nixon, who preferred the risk of conviction and removal to its virtual certainty. Thus, the president would prefer to defy a weak decision (outcome II in figure 6.1) than to defy a unanimous decision (outcome III), which I indicate by II ≻ III. For the same reason, Nixon would prefer to comply with any adverse decision (I or IV) than to defy a unanimous decision (III)— his worst outcome— so I, IV ≻ III.

Defying a weak decision (II) is considered preferable to complying with any adverse decision (I or IV), for such defiance would preclude the release of potentially devastating evidence and at the same time present Nixon with the possibility of avoiding conviction and removal for noncompliance; hence, II ≻ I, IV. Between the two compliance outcomes (I and IV), I assume that the president preferred to comply with a weak decision (I) than a unanimous decision (IV), so I ≻ IV.

A weak decision with some justices dissenting would leave the issue confused and subject to interpretation; it would also leave room to maneuver for partial compliance.6 Putting the partial preference rankings together, the president’s ranking of the four outcomes is II ≻ I ≻ IV ≻ III.

I next turn to the rankings of Burger and Blackmun. Although I suggested earlier that Burger and Blackmun would prefer to decide for the president on at least one of the strictly legal questions (standing to sue by the special prosecutor), there is no doubt the that the justices believed (p.136) that compliance by the president with any adverse Court ruling (I or IV) would be preferable to defiance (II or III); hence, their partial ranking is I, IV ≻ II, III. Indeed, in the Court’s opinion, which Burger wrote but which actually was drafted by the other justices (Schwartz 1996, 147; Woodward and Armstrong 1979, 340–341), the chief justice quoted Chief Justice John Marshall in Marbury v. Madison (1803): “It is emphatically the province and duty of the Judicial Department to say what the law is.”7

It is plausible to assume that if the president complied, the justices would prefer to decide for him (I) rather than against him (IV); hence I ≻ IV. After all, the notion that the Court must be unanimous or close to it to make a decision credible, and thereby induce compliance, is an undesirable restriction of the Court’s authority and might establish an unhealthy precedent.

Finally, I assume that the justices preferred that the president defy a unanimous decision (III) than a weak decision (II)— to which his chances of eventual success would be higher— so III ≻ II. Putting the partial preference rankings together, the justices’ ranking of the four outcomes is I ≻ IV ≻ III ≻ II.

6.3 Analysis of the White House Tapes Game

Because the players in the White House tapes game did not make simultaneous choices in ignorance of each other, the payoff matrix in figure 6.1 is not an accurate representation of this game according to standard game theory (in note 8 I briefly comment on the application of TOM to this game). As in the Abraham-God game in section 2.2, however, the 2 × 2 game offers a convenient way to describe player preferences, even if it was not the game played.

In fact, Burger and Blackmun— and the rest of the Court— acted first. Only then did Nixon have to make a strategy choice, as shown in the 2 × 4 payoff matrix in figure 6.2, wherein Burger and Blackmun have two strategies whereas Nixon has four. This is because each of Nixon’s two original strategies, comply (C) and defy (D), are contingent (p.137)

Law: Supreme Court Challenges and Jury Selection

Figure 6.2 Revised payoff matrix of White House Tapes Game

on what Burger and Blackmun decide: for (F) or against (A) the president.

In the figure 6.2 game, tat-for-tit is a weakly dominant strategy for Nixon. While Burger and Blackmun do not have a dominant strategy, they can anticipate Nixon’s choice in a game of complete information. To maximize their payoff in the fourth column associated with Nixon’s weakly dominant strategy, Burger and Blackmun would choose A, because they prefer (3,2) to (1,4).

As already indicated, the Supreme Court did decide unanimously against President Nixon. Nixon was reportedly shocked by the Court’s ruling, feeling himself “sold out” by his three appointees, Chief Justice Burger and Associate Justices Blackmun and Powell. Charles Colson claimed that the president counted on all three justices. Others say he was certain only of Burger and Blackmun.

When he learned of the decision, Nixon used foul (“expletive-deleted”) language to describe Burger. The president could not believe that the (p.138) Court’s ruling had been unanimous. “Significantly, the President’s greatest fury seems to have been directed not at the decision itself but at the three Justices who ‘deserted’ him” (Lukas 1976, 519).

In any event, the decision was unanimous, with no dissenting or concurring opinions. “It was the Court’s seamless unity which made defiance so difficult” (Ibid.). Eight hours after the decision was handed down, the president, through St. Clair, announced his compliance with the decision “in all respects.”

The game-theoretic analysis explains well, in terms of the foregoing reconstruction of the players’ strategies and preferences for outcomes, why they acted as they did: (3,2) is the unique Nash equilibrium in the figure 6.2 matrix. Notice, however, that both players could have done better if they had chosen strategies associated with either of the two (4,3) outcomes in the 2 × 4 game.

Both of these Pareto-superior outcomes involve the choice by Burger and Blackmun of deciding for the president, and the choice by Nixon of compliance. The president can implement this Pareto-superior outcome by choosing either comply regardless or tit-for-tat.

Unfortunately for the players, however, neither of the outcomes that yield (4,3) is in equilibrium: Nixon in each case has an incentive to depart unilaterally from his strategies associated with (4,3) in order to bring about his best outcome, (1,4). Not only are the (4,3) outcomes not in equilibrium, but also Nixon’s strategies associated with these outcomes are weakly dominated by his strategy of tat-for-tit.

For these reasons, therefore, it is hard to see how both players could have done better, even though the opportunity existed. Only if Burger and Blackmun had believed that their dissent would not trigger presidential defiance could they have voted their convictions with greater equanimity. The public record, however, shows that Burger and Blackmun never received any such assurance.

Quite the contrary: Nixon and his spokesmen, as indicated earlier, continually held out the possibility of defying a Supreme Court decision that was not “definitive.” Thus, Burger and Blackmun had no choice— despite their disagreement with some arguments of the special prosecutor— but to decide against the president. Thereby the Supreme Court decision was rendered unanimous and both players in the White House tapes game lost out, in a sense, on greater payoffs that, at least in principle, were attainable.

(p.139) The public probably gained from this outcome, however. If one identifies the public with the special prosecutor, then the game that the special prosecutor set up, though he himself was not a player, yielded the public, at (3,2), probably its best possible outcome of the four that were possible. This is certainly a reasonable inference from Jaworski’s remarks immediately after the Court decision: “I feel right good over what happened. We can move ahead now.…I’m especially pleased it was a unanimous decision. It doesn’t leave any doubt in anyone’s mind” (New York Times, July 25, 1974, 22).

It is worth pointing out that a variety of bizarre motives (need to fail, death wish) and personality traits (self-destructive) have been attributed to Richard Nixon. The analysis here, however, suggests that his stance in the White House tapes case, which pushed his confrontation with the special prosecutor and then the Supreme Court beyond the point of no return, was not at all strange. Rather, Nixon was caught up in an intractable game— partly of his own making— that, perhaps with greater prescience, he could have avoided.8

After the Court rendered its verdict, Nixon gave an eminently sensible reason for resigning: “I no longer have a strong enough political base [to complete the term of office]” (New York Times staff 1974, vii). This rationale is as good as his reason for obeying the Supreme Court’s (p.140) edict— that he could not do better by defiance. Complex and enigmatic as Nixon was, he was, at root, a rational player.

6.4 The Roosevelt Court and the New Deal

Franklin D. Roosevelt was inaugurated as president on March 4, 1933.9 Less than a year earlier, on March 14, 1932, Benjamin Cardozo had been sworn in as associate justice of the Supreme Court, which rounded out the group of justices that would preside over Roosevelt’s first term as president.

Cardozo quickly allied with the Court’s two liberal associate justices, Louis Brandeis and Harlan Stone. Together, this trio formed a bloc that would be almost completely unshakable for the next six years on a Supreme Court led by Chief Justice Charles Evans Hughes, a former governor of New York (1907–1910), associate justice of the Supreme Court (1910–1916), Republican presidential candidate (1916), and secretary of state (1921–1925).

On the opposite end of the Court’s ideological spectrum was a conservative bloc of four associate justices, who were known as the Four Horsemen for their unmerciful attacks on President Roosevelt’s New Deal legislation. This quartet comprised Willis Van Devanter, George Sutherland, Pierce Butler, and the infamously sour James McReynolds, the most implacable opponent of the New Deal and also a notorious anti-Semite (Shesol 2010, 102).

In the center of the spectrum, playing moderators between the two often-warring factions, was Hughes and the incomprehensibly unpredictable Associate Justice Owen Roberts. In effect, the Court had four independent players, the Trio, the Horsemen, Hughes, and Roberts.

As Roosevelt’s New Deal legislation came before the Court, these players had to decide on the constitutionality of the progressive laws that had been passed by Roosevelt’s Democratic Party-controlled Congress. In the rows of figure 6.3, I show all possible winning coalitions on the Supreme Court, with at least five members, who can rule on the constitutionality of these laws and so determine their fate. (p.141)

Law: Supreme Court Challenges and Jury Selection

Figure 6.3 Critical defections and Banzhaf power of players

In each row of figure 6.3, the ×s indicate the players who are critical in each coalition: By defecting from a winning coalition, each would cause it to be losing. For example, in the first row, if any one of the Trio, Hughes, or Roberts defected, the five-person winning coalition would be reduced to losing status. By contrast, in the fifth row, no defection of any player from the grand coalition of all the players would cause it to be losing.

Counting up the ×s in each column, and dividing by the total number of ×s of all the players (twelve), gives the percent of voting power of each player. This measure of relative power is called the Banzhaf index (Banzhaf 1965). It has been applied to many voting bodies with differently weighted players.10

(p.142) Figure 6.3 shows that the Horsemen, whose votes are critical in six of the eight winning coalitions, have 1/2 the voting power. Compared with the Trio— with only one less vote— the Horsemen have three times as much voting power (1/2 versus 1/6). In fact, the Trio has exactly the same voting power as each of the single moderates, Hughes and Roberts.

Does this index provide an accurate reflection of the distribution of power on the Court? The answer is no, because it supposes that all conceivable winning coalitions of the four players are equally likely, which was decidedly not the case.

In fact, the Trio never coalesced with the Horsemen, with or without the moderates, in cases that involved New Deal legislation. In the fifteen such cases that the Court heard between 1934 and 1937, none was decided by any of the coalitions that are crossed out in figure 6.4.

Law: Supreme Court Challenges and Jury Selection

Figure 6.4 Critical defections and Banzhaf power of players with restriction

(p.143) With these ×s eliminated, the distribution of voting power changes radically. While the Horsemen drop from 1/2 to 3/8 and the Trio from 1/6 to 1/8, Hughes and Roberts each go from 1/6 to 1/4, increasing their voting power by 50 percent. Astonishingly, each of the moderates is now twice as powerful as the Trio, even though the Trio has three times as many votes. Additionally, the sum of the Banzhaf powers of the two moderates (1/2) is greater than the Banzhaf power of the four Horsemen (3/8), which is also the power of moderates acting as a bloc.11

The power of the moderates derives from their being swing justices, each able to make the Horsemen winning and, together, being able to make the Trio winning. In fact, the Banzhaf attribution of power to the moderates, given the restriction on coalitions that can form, echoes well what happened on the Court. When the moderates became the decisive players, President Roosevelt bitterly complained at his weekly press conferences that the economic stability of the nation rode on the whims of Hughes and Roberts (McKenna 2002, 217).

From early 1934 until Roosevelt announced his Judiciary Reorganization Bill in 1937, the Court rendered ten divided cases, relating to the New Deal legislation, by margins of 5–4, 6–3, and 7–2. Roberts voted with the majority in every one— in fact, eight were decided by his vote— siding with the Trio five times and the Horsemen five times. He often articulated sweeping legal precedents in his opinions, sometimes voting against decisions that he had supported only months earlier.

Chief Justice Hughes was hardly more consistent. He voted with the Trio in all of the New Deal cases in 1934 and 1935, but then he ended up voting with the Horsemen in half of the nonunanimous cases of 1936 and the Trio in the other half.

By mid-1936, Roosevelt was infuriated at the see-sawing decisions of the Court. After winning a resounding victory for a second term on November 3, 1936— receiving 61 percent of the popular vote and more electoral votes than had ever been won in a presidential election— he decided to try to gain more decisive control over the erratic decisions of the Court.

(p.144) Emboldened by his election triumph, he announced his plan to “pack” the Supreme Court on February 5, 1937, soon after he was inaugurated. The Judiciary Reorganization Bill of 1937 (a.k.a “JRB” or “the court-packing plan”) would allow the president to appoint a new Supreme Court justice, up to a maximum of six, whenever a current justice over the age of 70 had not retired.

For Roosevelt, this would mean the appointment of six new justices, increasing the size of the Court from nine to fifteen justices. A public outcry against Roosevelt’s plan ensued, however, causing the defeat of JRB by 70–20 in a Senate vote on July 22, 1937. But the threat to pack the Court had, arguably, already compelled it “to assume a more liberal outlook” (Shesol 2010, 502).

In calculating Banzhaf voting power, so far I have assumed that Hughes and Roberts act independently. Would they do better if they formed a two-person moderate bloc (Duo) that always voted together— assuming they can vote either with the Trio or the Horsemen, but the Trio and Horsemen can never vote with each other?12

Now there are only two winning coalitions that are possible: (Duo, Horsemen) and (Duo, Trio). Because the Duo is critical in two and the Horsemen and Trio in one each, the Banzhaf power of Duo is 1/2 and of Horsemen and Trio 1/4 each. Paradoxically, this gives the smallest player power equal to the two larger players combined. Does this attribution reflect what actually happened?

It is instructive to look at both the goals and voting records of Hughes and Roberts. Hughes’s primary goal was to maintain the integrity of the Supreme Court. He thought that the authority of the Supreme Court rested on its ability to act cohesively, and he hated handing down 5–4 decisions (Solomon 2009, 49). Voting based on his personal ideology was only a secondary consideration.

(p.145) It is more difficult to ascertain Roberts’s preferences. He liked to present himself as a man of the law and frequently based his vote on legal technicalities. At the same time, he claimed that he approached each case based on its merits and the facts. He said he did not vote along ideological lines, which is a claim that for a time seemed supported by his unpredictable voting record.

Roberts enjoyed spending his time away from Supreme Court work wandering the Capitol grounds, “chatting with the messengers and tourists” (Solomon 2009, 63). He refused to eat his lunch in the Senate cafeteria, dining instead in public places. He fancied himself a man of the people and their advocate on the Court.

After Roosevelt’s landslide victory in 1936, Roberts believed that the people had spoken forcefully. Three months after the election, and a mere eight months after ruling a New York state minimum-wage law to be unconstitutional, Roberts reversed himself and voted to uphold a near identical Washington state minimum-wage law in West Coast Hotel Co.v. Parrish (1937), casting his vote before the court-packing plan was even announced (McKenna 2002, 418). In effect, Roberts realigned his jurisprudence with the opinion of the citizenry, at least as he interpreted it (Shesol 2010, 405–408, 412–415).

Between March and May of 1937, ten New Deal cases came before the Court, and the Court upheld the legislation’s constitutionality in every one. Both Hughes and Roberts voted with the Trio in each, eight of which were decided by a 5–4 vote. In effect, the Trio turned into a New Deal Quintet. Thereby the divided Court swung decisively in favor of the liberals, mirroring the voting power that the Banzhaf index attributes to the moderate Duo when coalitions are restricted in the manner described.

The analysis of this section illustrates another application of cooperative game theory, which was introduced in chapter 4. As in the analysis of the fair division, the question is how to divide something of value (indivisible goods in the case of the fair-division application, voting power in the case of players of different weight). Unlike the earlier analysis, however, this chapter demonstrates how a theoretical model elucidates an empirical case instead of asking what theoretical properties are compatible and normatively desirable.

(p.146) 6.5 Jury Selection

There is probably no right more basic in the western legal tradition than that of trial by jury.13 It is a right guaranteed by the U.S. Constitution in both civil and criminal cases, although the Constitution leaves undefined the procedure by which a jury is to be selected. Nevertheless, the right of the defense and prosecution to challenge prospective jurors— called venirepersons (venire is Latin for “to come”), who are summoned to jury duty— has a long history in criminal law, going back at least to Roman tribunals (Abraham 1975, 116).

In the United States, challenges exercised in the selection of a jury are of two types, for cause and peremptory. In the case of the former, the challenger must advance a bona fide reason why a venireperson is unacceptable, and therefore should be disqualified from service by the court, because of bias, prejudice, or other involvement in the substantive or procedural aspects of the litigation that could affect his or her judgment in reaching a verdict. Because there are no limits on the number of challenges for cause, this type of challenge raises no strategic questions of timing.

In Swain v. Alabama (1965), the Supreme Court considered whether peremptory challenges by the prosecution could be used to exclude all blacks from a jury. Upholding peremptories as an absolute and unchallengeable right, the Court said, in a 6–3 decision written by Associate Justice Byron White, that “the function of the [peremptory] challenge is not only to eliminate extremes of partiality on both sides, but to assure the parties the that jurors selected before whom they try the case will decide on the evidence placed before them, and not otherwise.”

One implication of this statement is that an impartial jury is one in which the “extremes of partiality” have been eliminated on both sides insofar as possible. Given this interpretation, define the partiality of a venireperson as his or her probability of voting for conviction in a criminal trial before hearing the evidence. That is, before the trial commences (p.147) and evidence is presented, assume that each venireperson can be assigned a probability of voting for conviction. Then define, consistent with the Supreme Court ruling, an impartial jury to be one in which those venirepersons with the highest and lowest probabilities are peremptorily challenged by defense and prosecution— assuming excusals for cause have already been rendered— leaving a jury whose members have probabilities in some middle range.

For most jury-selection procedures presently in use, there is no way that the defense and prosecution can exercise their peremptories so as to ensure that one venireperson, selected as a juror, has a less extreme probability than another one not selected to serve. This is because under all procedures except the “struck jury system,” peremptories are exercised by each side over time. As the selection process proceeds, one cannot know for sure who the most extreme venirepersons will be and challenge only them.

Jury selection is very different under the struck jury system, wherein peremptory challenges are exercised just once after the examination of a panel of venirepersons equal in size to the sum of the number of jurors to hear the case plus the number of peremptory challenges each side is allowed. In fact, only under this system can one assuredly eliminate all venirepersons— up to the limit of one’s peremptories— whom one views to be least favorably disposed to one’s side. Since one does not have to worry that one will let pass a more extreme venireperson— or challenge a less extreme venireperson— who appears later in the sequence, the most extreme venirpersons can be removed in one fell swoop. If neither side challenges the same venireperson, a jury of requisite size will be chosen.

But what if challenges are sequential, and the prosecution or defense attorneys must decide, immediately after the questioning of a venireperson in the so-called voir dire— literally, “to speak the truth”— whether or not to challenge him or her?14 This decision cannot be made lightly, because attorneys have a limited number of peremptory challenges to exercise.

More specifically, federal law provides for twenty challenges when the punishment may be death, ten for other felonies, and three for (p.148) misdemeanors. Statutes in most states provide for a similar distribution, depending on the seriousness of the alleged crime, with no state providing for less than two or more than thirty peremptory challenges. In civil trials, the number of peremptories allotted to each side is generally less than in criminal trials (McCart 1965, 33).

In the jury-selection game, each venireperson comes up one at a time for consideration after the voir dire. I assume that both sides agree on his or her probability of voting for conviction, which may vary between a low of 0 and a high of 1. At least for the most extreme venirepersons with very low or very high probabilities, it is reasonable to assume that a venireperson the defense considers most favorable (low probability of convicting) the prosecution will view as most favorable (high probability of convicting), and vice versa.

This view of the game receives support from a report by social scientists working for the defense in the “political” trial of the Harrisburg Seven in 1972 (Schulman et al. 1973). The jury-selection procedure used in this trial was that of panel challenges, whereby a panel of venirepersons was first chosen before peremptory challenges were exercised to reduce it to a jury of twelve members; hence, the one-at-a time assumption is inapplicable. Nonetheless, it is noteworthy that the prosecution applied all six of its peremptory challenges to the eight members of the forty-six-member panel that the defense rated 1 (best) on a five-point scale, thereby eliminating all except two of the defense’s choices. If challenges had been exercised randomly by the prosecution against the fortysix-member panel, the chance that it would apply all its peremptories to the defense’s eight best choices is less than one chance in 300,000 (Brams and Davis 1978, 976).

A second assumption of the jury-selection game is that both sides know how the probabilities of voting for conviction are distributed in the population from which venirepersons are chosen. They know, for example, whether most venirepersons have low probabilities, high probabilities, or are evenly distributed across the spectrum.

Indeed, in the Harrisburg Seven trial, the defense made a major effort before the start of the trial to survey the community from which the jurors would be drawn and learn something about its attitudes relevant to the defense’s case, as well as their correlation with certain demographic factors like age, education, and occupation. This enabled the (p.149) defense to compare venirepersons against the statistical profile of a favorable juror.

Michael J. Saks (1976, 15), a skeptic of such selection procedures for weeding out unfavorable jurors, argues that “trial evidence is and always has been a far more important determinant of the verdict than who sits on the jury.” However, he admits that “if the evidence is close, then scientific jury selection could make the difference,” adding that

when the same information is available to a human decisionmaker and a mathematical model, almost without exception the mathematical model makes more reliable predictions. After sixty studies comparing clinical versus statistical prediction, the human beat the computer only once. (Ibid.)

A third assumption of the game is that the venirepersons selected as jurors will independently reach a verdict according to their probabilities of voting for conviction. This assumption, of course, ignores the twin effects of evidence presented in the trial and jurors’ influencing each other in their deliberations in the jury room. These effects notwithstanding, the model does give insight into the selection of a jury before the start of a trial, which is important if its composition is likely to influence the outcome.

Insofar as this is true, the selection of an “impartial jury” thus becomes an exercise in rejecting the most partial jurors— as scientifically as possible— rather than a high-minded attempt to define and choose fair and impartial jurors. Jurors are selected because they are viewed as favorable to one’s case, not because they are considered paragons of fairness.

The forgoing assumptions of the jury-selection game allow one to calculate recursively the threshold probabilities of conviction above which the defense should challenge and below which the prosecution should challenge a venireperson at each point in the selection of a jury. To illustrate how the results of this calculation can be applied, assume that the probabilities of voting for conviction in the population of venirepersons from which jurors are chosen are uniformly distributed: Venirepersons who come up for examination are equally likely to have low, moderate, or high probabilities of voting for conviction.

Assume also that the defense and prosecution each have ten peremptories, and a jury of twelve members is to be chosen that must reach a (p.150) unanimous decision. At the beginning of the game, when no jurors have yet been chosen, the model prescribes that the defense should challenge a venireperson if he or she has an a priori probability of voting for conviction greater than 0.742, and the prosecution should challenge a venireperson if he or she has an a priori probability of less than 0.407.

Thus, the defense should choose a more “conservative” strategy by challenging venirepersons closer to its extreme of 1 (over the smaller range of 0.742 to 1.000) than the prosecution should by challenging venirepersons farther from its extreme of 0 (over the larger of range of 0 to 0.407). Intuitively, the defense can afford to be more conservative in eliminating venirepersons, because it needs only one vote to prevent conviction of the defendant.

What are the optimal strategies of both players near the end of such a game? Assume eleven jurors have already been chosen, and each side has just one peremptory challenge remaining. Then the defense should now challenge a venireperson with a probability greater than 0.625, and the prosecution should challenge a venireperson with a probability less than 0.375. Thus, the defense should now move farther from its extreme of 1 than previously, and the prosecution should move closer to its extreme of 0, so that now both sides challenge an equal distance from their extremes of 1 and 0.

Similar examples of optimal strategies can be calculated for any stage in the jury-selection game. The two sides need not have equal numbers of peremptories remaining— nor start with equal numbers— nor does the jury have to comprise twelve members. Neither does one need to assume that the probabilities of venirepersons are uniformly distributed; they could, for example, be concentrated in the middle near 0.5, according to a bell-shaped curve, or be skewed negatively (more concentrated on the right toward the prosecution) or positively (more concentrated on the left toward the defense).

The strategies that I have illustrated are optimal in the sense that they minimize for the defense, and maximize for the prosecution, the expected probability of conviction in a two-person constant-sum game between prosecution and defense. Given the assumptions of the model, if either side deviates from these optimal strategies, it will be hurt if the other side plays optimally, rendering these strategies a Nash equilibrium.

(p.151) The model can be modified to take account of challenges to panels of members. If, for example a panel of twelve venirepersons is first examined before each side can peremptorily challenge any of its members, and those challenged are replaced— and so on, until a jury of twelve unchallenged members is chosen— optimal strategies can be derived, but they are more complicated than those described earlier. Specifically, they are mixed rather than pure strategies, which is to say that both sides must make random choices to hide from the other side exactly whom they challenge at each stage in the game (though this information would become known at the next stage).

It is not necessary to model all extant peremptory-challenge procedures that occur in stages, and derive optimal challenge strategies for each, to prove that they cannot ensure an impartial jury. As I indicated earlier, any procedure that requires that challenges be made sequentially, rather than all at once, allows for the possible selection of more extreme venirepersons than those who are challenged and thereby removed.

One can illustrate this point with the earlier example. Suppose that a venireperson with a probability less than 0.375 comes up near the end of the game, when eleven jurors have already been chosen and each side has one peremptory remaining, so the model prescribes that he or she be rejected by the prosecution. Now since the prosecution has no peremptories remaining after challenging this venireperson, it must accept the next venireperson to come up (if the defense does), even if he or she has a probability of 0 of voting for conviction. Clearly, choosing optimal strategies in the game does not guarantee that the “extremes of partiality,” in Justice White’s words, will always be eliminated.

The reason for constructing the jury-selection model is that it enables one to establish a challenge threshold as optimal, even in the face of uncertainty. Not knowing who will come up next, but only the distribution from which he or she will be chosen, it tells the prosecution in the foregoing example that it will do worse, on average, by choosing a challenge threshold other than 0.375. It is this threshold, and only this threshold, that maximizes the jury’s expected probability of voting for conviction at the particular stage in the game described. But even adopting optimal strategies, the defense and prosecution may not succeed in obtaining impartial juries— as judged by the “extremes of partiality” criterion—because (p.152) what is best on average may not be best, ex post facto, in a particular case.

Because trial lawyers must play averages in an uncertain world, the optimal challenge threshold given by the model may be useful, given that its assumptions reasonably approximate actual jury-selection procedures. But the strategic calculations raise a more fundamental question than that of what strategy is best. Because what is best depends on the rules, the question of whether different rules might better enable both sides to eliminate the extremes, and thereby obtain a more impartial jury, is also important to consider.

As I showed earlier, there is no way to guarantee the elimination of these extremes unless peremptories are exercised against all venirepersons at once under a struck jury system. Arguably, this system should replace all other procedures currently governing the exercise of peremptory challenges in trials today. This reform would shift the question of when and whom to challenge at each stage to, more simply, whom to challenge in a single stage.

Such a reform seems amply justified by the constitutional guarantee of an “impartial jury” explicitly provided for by the Sixth Amendment “in all criminal prosecutions.” Even in civil cases covered by the Seventh Amendment, in which the word impartial is not used, the language (“the right of trial by jury shall be preserved…according to the rules of the common law”) conveys, at least implicitly, the spirit of impartiality.

I conclude that all peremptory-challenge procedures except the struck jury system are well supported by both the Constitution and the Supreme Court’s interpretation of what constitutes an impartial jury. In fact, the cause of equal justice under the law is best served by a procedure that renders strategic calculations irrelevant and hence affords no advantage to the better strategist. There is no constitutional justification, or other legal basis of which I am aware, for retaining jury-selection procedures that encourage strategic calculations and choices that, even if optimal, may distort the selection of an impartial jury.

6.6 Summary and Conclusions

This chapter has progressively gone back in time. I began with a 1974 presidential challenge to the authority of the Supreme Court that did not, (p.153) in the end, materialize. I then moved back to a 1937 presidential challenge that also failed. I concluded by exploring how an impartial jury, a notion that is embodied in the U.S. Constitution that was ratified in 1789, could be achieved.

The conflicts between Presidents Nixon and Roosevelt and their Supreme Courts have the drama of a novel; the protagonists make calculations, sometimes flawed, which game theory helps to explicate. Game theory also helps to illuminate the calculations that prosecution and defense make in the selection of an impartial jury and identifies the procedure that would best lead to this ideal.

Richard Nixon, by implicitly threatening not to abide by a decision of the Supreme Court if it were not definitive, was thwarted in his attempt to save his presidency. His threat caused two of his Supreme Court appointees, who thought Nixon had grounds to withhold the tapes, to coalesce with the majority— who were disposed to order the release of the tapes— in order to put up a united front that Nixon would not be able to defy. After the tapes incriminated Nixon in a scheme to cover up a crime, he was forced to resign, though he was later pardoned by President Gerald R. Ford, his vice president who succeeded to the presidency.

Franklin Roosevelt was similarly thwarted by a Supreme Court that was not always willing to uphold his New Deal legislation. In this conflict, the game theory analysis shows how the two members of the Court who could decide which way the Court swung had their power greatly enhanced.

Insofar as these justices acted as a bloc, they had as much power as the other seven justices combined (three liberals, four conservatives), which vividly illustrates how the voting power of blocs, at least as measured by the Banzhaf index, may have little relationship to the numbers of votes controlled by the blocs. As the analysis showed, the ability of players to change outcomes is also a function of where these blocs are situated on the ideological spectrum and, therefore, with which other blocs they will coalesce to form a majority.

The selection of juries has long seemed more an art than a science. Beginning with political trials of dissidents in the 1970s, however, jury selection became more scientific.A game theory model shows, given that both sides can judge the favorableness of prospective jurors, how one (p.154) can determine thresholds that indicate when these venirepersons should be accepted or challenged.

These calculations are not surefire, however, if one must make them sequentially. Consequently, I argued for a struck jury system, in which such calculations are rendered irrelevant because the peremptory challenges are exercised all at once.

To conclude this chapter, it is worth asking whether law can properly be classified as a humanities discipline. After all, one finds only a scattering of law-related courses taught in humanities fields like history or philosophy. Like medicine, law is first and foremost a profession, taught in law schools.

The inclusion of law in this book is justified, in my view, by the fact that the players in law-related games make fundamentally human choices. Emotions, especially anger, which I analyze more formally in later chapters, are front and center in the Nixon and Roosevelt games. And while I show that jury selection can be mathematized to a degree, the input to the game theory model requires human judgments about the leanings of prospective jurors, which cannot so easily be mathematized.

In short, law is built on the strategic choices made by people in real-life situations; they are not intrinsically different from the cast of characters who populate novels and plays, which Posner (1998) explicates with many examples from the literary canon. Like their fictional counterparts, real characters, when constrained by laws, play games. They just do so more in the shadow of the law than their fictional counterparts.15


(1.) This and the next section are adapted from Brams and Muzzio 1977a, 1977b with permission; see also Muzzio 1982, chap. 3, and Brams 1985, chap. 7.

(2.) Associate Justice William Rehnquist (who later became the chief justice) withdrew from the case, evidently because of his previous service in the Justice Department under Attorney General John Mitchell, though he never publicly stated a reason for disqualifying himself.

(3.) Data on case agreement can be found in the November issues of the Harvard Law Review (1971–1975). On the concurrence of Burger and Blackmun, see the New York Times, July 1, 1974, 10; and Totenberg 1975.

(4.) I do not include Powell as a Court player, because he originally favored granting certiorari; also, he “demonstrated the highest level of independence within the Nixon Bloc” and was described as “one of the least predictable of the eight and most flexible of the Nixon appointees” (New York Times, July 1, 1974, 10; Time, July 22, 1974, 16).

(5.) This view is corroborated by H. R. Haldeman: “If the Supreme Court had handed down a [nonunanimous] majority decision, Nixon would have defied the Court and refused its order to turn over the tapes” (Haldeman and DiMona 1978, 310).

(6.) It can reasonably be argued that the president preferred to comply with a unanimous decision (IV) than a “nondefinitive” ruling that he had been threatening to ignore (I), so IV ≻ I. The reversal of the ranking of the two compliance outcomes leads to essentially the same results that I will subsequently describe, except that the Nash equilibrium becomes (3,3) rather than (3,2).

(7.) Marbury is analyzed as a two-person game between President Thomas Jefferson and Chief Justice John Marshall in Clinton 1994.

(8.) So how might the Pareto-superior (4,3) outcome have been induced? According to TOM (section 2.5), (4,3) is the unique nonmyopic equilibrium (NME) in the 2 × 2 game in figure 6.1, so wherever play starts, farsighted players would ultimately choose it. In addition, the exercise of moving power (section 3.4)— whichever player possesses it— reinforces the choice of (4,3). Finally, the exercise of threat power (to be discussed in section 7.2) by either player also leads to its choice. Perhaps the best TOM-based explanation of why (4,3) was not chosen is that Burger and Blackmun were not capable of threatening Nixon with the possibility of a unanimous decision before they had to decide, which complements the explanation given in the text— Burger and Blackmun’s choice in the 2 × 4 game in figure 6.2 precedes Nixon’s choice, so they cannot signal Nixon of their intent before acting. In retrospect, Nixon was foolish to threaten the Court with the possibility of defiance; anticipating this choice in the figure 6.1 game, Burger and Blackmun rationally responded by voting against the president, because they preferred (2,1) to (1,4). To be sure, had Nixon been more reassuring that he would comply with any Supreme Court decision, including a divided one, then he would have been in deep trouble if Burger and Blackmun had chosen F and made the decision 6–2. For then he would have had to backtrack on his promise to comply, which almost surely would have led to his impeachment and conviction. In short, Nixon was in a no-win situation, whatever he said about the Supreme Court’s impending decision. A TOM analysis of the strategic interaction over several decades between the Israeli Supreme Court— a more powerful court than the U.S. Supreme Court— and the Knesset (the Israeli parliament) is given in Meydani and Mizrahi 2010.

(9.) I am grateful to Edward W. Larroca for giving me permission to adapt material from his student paper for this section.

(10.) For an overview of the Banzhaf and related power indices, their properties, and their applications, see Felsenthal and Machover 1998.

(11.) Later in this section I show that if the moderates form their own bloc, their power is equal to that of the Trio and Horsemen combined, again assuming the latter two players cannot be members of the same winning coalition.

(12.) Schubert 1958 also treated Hughes and Roberts (whom he called “Hughberts”) as a single player but used a different power index (that of Shapley and Shubik 1954) and made somewhat different assumptions about coalitional alignments that were feasible. He showed that each of the three blocs had approximately equal power, determined empirically, unlike the finding discussed next in the text that the Duo has the same power as Trio and Horsemen combined. For a review of this and other early applications of game theory to the Supreme Court, see Brenner 1980.

(13.) This section is adapted from Brams and Davis 1976, 1978 with permission; a very similar model of jury selection was developed independently by Roth, Kadane, and DeGroot 1977. Related models of sequential selection are analyzed and compared in Alpern, Gal, and Solan 2010 and citations therein.

(14.) While I will speak of “prosecution” in the subsequent discussion, one can substitute “plaintiff” for the side that initiates the case if the trial is civil.

(15.) Law serves as background to the situations described in Kaminski 2004, whose author demonstrates, using game theory, how criminals incarcerated in Polish prisons strategize.