Of Love, Lust and Perjury A case study When Monica told her best friend Linda Tripp of her love for the President, she soon realized that with friends like Linda, she did not need enemies. Playful rituals in the Oval Office were revealed to a public, hungry for the juicy details of sexual misbehavior. Bill, the mature if still green President, had engaged in foreplay, and perhaps even sex, with a young intern. But Bill was going to hedge. He did not have sex with that woman, whatever sex meant. Devastated and probably heartbroken Monica went to Mom, the only person she could trust. And Mom made sure to save the evidence: she put the navy blue dress, unlaundered, in her New York apartment. Ken Starr did not know how much a mysterious blue dress would have to say when he weighed granting Monica immunity against the threat of punishing her for lying about her relationship with Bill. And neither did Bill. Only Monica had some idea of what the dress could reveal, but wondered whether it was really necessary to say the whole truth if what was said was no less than truthful. After all, this was not just about sex. There was affection too and perhaps some hope that Bill really did love her, and would own up to a passionate hug, instead of claiming a seedy proximity whose sexual overtones did not amount to love making. Bill, meanwhile, facing dismissal from the Presidency for a frolic, was tempted to discredit Monica s story as the teenage fantasy of a Cosmo girl. 1
Game Structure and Payoffs What can Bill, Monica and Ken learn from game theory? Should Bill and Monica tell the truth, the whole truth and nothing but? Or should they stick to an incomplete version of the story. Should Bill discredit Monica? Should Ken give Monica immunity, or should he count on the threat of dire consequences if she does not tell him everything? Our three actors, Ken the prosecutor, Monica the witness and Bill the suspect must think through the consequences of their decisions in view of what the others might want to do. Ken the Prosecutor would really like to catch Bill lying under oath. That, more than anything would make his day. Monica could help if she were to tell the truth, and the threat of imprisonment for the next twenty seven years might well encourage Monica to lay her life story at Ken s feet. But immunity might encourage even greater cooperation especially if the hapless intern has been scared out of her wits by Starr deputies in a Crystal City hotel room. 1 Would Monica be willing to turn over a certain blue dress if given immunity? And if dress there was, thought Ken, it could prove to be Bill s undoing but he could not know for sure. If there was definitive proof out there of Bill s misdeeds, Ken reasoned, then Monica was aware of it and might hand it over to him if given immunity. And Bill did not know whether the dress would be there to incriminate him and could be tempted to lie. Ken s ideal scenario would then come to pass and this would be his motivation to give Monica immunity. But what if Ken gave Monica immunity and the dress did not yield conclusive evidence? Ken would have lost the prison threat weapon for nothing and Monica could now lie to protect her lover. But then, Ken 1 On October 23, 1998, Lars Erik Nelson reported in the Daily News that Starr deputies had threatened Monica Lewinsky with 27 years in prison for perjury and obstruction of justice for having signed an affidavit claiming that she had had no sexual relationship with the President. (How Starr Trapped Clinton, October 23, 1998, p. 59). According to Nelson, Monica had not committed perjury at the time because the affidavit in question had not been filed. 2
reasoned, a lover spurned is a most ruthless enemy. Monica could well say all to spite the man whose throne comes first, before the love she thought they shared. And faced with Monica s story Bill might be tempted to tell the truth just in case the dress could be revealing. After all, this was not a bad outcome although catching a perjured Bill redhanded would be more satisfying. So we begin construction of our game tree with Ken s ruminations: He must decide whether to give Monica immunity or not without knowing whether he can get his hands on definite evidence of the extent of Bill and Monica s relationship: Ken s decision to offer or to deny immunity to his star witness is made with uncertainty about the quality (hard or soft) of the evidence against Bill. Ken therefore has two decision nodes, one for each circumstance, and they belong to an information set. In other words, Ken does not know whether he will be at node immunity(h) or immunity(s) when he makes his decision. But he can have expectations about the probability with which he could find himself in one position of the other. Here we have assumed that Ken believes that there is 60% chance that Monica will have hard evidence to deliver. His moves, to 3
deny or to offer immunity, are of course the same at each of the nodes in his information set. Each of these moves must lead to decisions by one of the other protagonists or be final. In this version of the perjury game we will assume that if Ken denies Monica immunity, she will tell most of the truth, Bill will lie and the dress will get laundered. Deny is a final move and we will have to figure out payoffs to each player if Ken decides to go that route. Monica the Witness: Monica is scared and confused and hurt. But she still likes Bill, the handsome Head of State whose murmured endearments in the Oval halls of power made her feel important, beautiful and desired. With no immunity she has to tell the truth but the dress can find its way to the nearest Laundromat. If Monica is to hand over the key to Bill s demise, immunity is the price that Ken must pay. And Monica secretly hopes that Bill will own up instead of sinking his own ship by defacing hers. When her turn comes on the witness stand, she must decide to tell the truth or to lie. Game theoretic integrity requires that she think through her decision whether Ken has hard evidence or not. But she reasons things out knowing that, if she accepts to bear witness with immunity, Ken will be fully armed. 4
If she decides to lie, Monica reasons, she could be charged with perjury if Ken has hard evidence to show. If not, then there s precious little that Ken can use against her, or Bill for that matter. Lying is a final move for this game and its payoffs to Ken, Monica and Bill depend, of course, on the nature of the evidence. If Monica plans to tell the truth, it will be Bill s turn. Bill the Suspect: Bill wants the whole affair to go away. But it won t. If Ken threatens Monica, she may not bear the strain and could blurt out the truth. She could do that too if she is given immunity. Then, perhaps it would be wise to just deny, and to accuse the poor girl of being delusional. But, in the back of his mind Bill has a doubt. Rumor of an incriminating dress has been floating around news rooms. He must think through the consequences of his decision to admit to the affair, or to discredit Monica without knowing how hard the evidence is against him. Like Ken, his decision nodes (defense(h) 5
and defense(s)) belong to an information set. Whatever he decides must apply at either of these nodes since he does not know where he will be. What he will know by then is that Monica has spilled the beans. To discredit or not will be the question. But whatever he does the game will close with his statement. Reward or punishment will be metered out. Both of Bill s moves are final. All players must eventually face the consequences of their actions. In fact it is the careful anticipation of these consequences that will determine each character s expectation of what the others will do as well as the decision that he or she will make in 6
response. All game-ending moves must come with payoff consequences for each of the players. In the perjury game, what thrills the prosecutor makes the suspect unhappy. Ken s best outcome is Bill s worst. And witnesses, by and large would rather tell the truth than to lie, especially if they know that hard evidence is available. There are certainly many ways to assign payoffs to the eight final moves of the perjury game, and it will be one of the objectives of sensitivity analysis to identify critical payoff assumptions. Critical payoff assumptions change the solution of the game when they are relaxed. The same applies to assumptions made about the structure of the game. But we are getting ahead of ourselves. Let s decide on a first set of payoffs for our players: Ken s preferred outcome is to catch Bill with his pants down: this happens if he gives Monica immunity, she has hard evidence, tells the truth and Bill tries to discredit Monica. Next best is to watch Bill admit, after having publicly denied any relation with that woman. Worst case scenario, Monica denies everything in order to protect her lover and Bill is off the hook. But this would only happen if there was no hard evidence against Bill. If Ken does not give Monica immunity, she will definitely tell the truth but Ken will have no hard evidence with which to trap Bill. Contradictions in the lovers stories might be useful weapons but would never match the weight of genetic revelations. If there was no dress to gain from immunity, Ken would lose less from denying Monica legal security than if a hard line towards the witness meant that he lost access to hard evidence. This reasoning determines Ken s relative payoffs if he denies Monica immunity Monica would rather tell the truth than lie especially if she has immunity. Having spilled the beans, she would like Bill to admit to the relationship without having to incriminate 7
him with a dress. But since blue dress there is, her next best outcome is to have Bill admit and face the evidence. Monica does not want to lie. And lying just to watch Bill get caught is almost as bad as having him deny her story. If she does not have immunity, she will tell them almost everything and Bill will discredit her, but Ken won t have the dress. This determines Monica s preference for the situation she would face if she did not have immunity. She prefers to be scared and to save Bill, than to be discredited while Bill falls for the perjury trap. Bill is happy to discredit Monica if it gets him off the hook. He would prefer to have her lie as well. Two is company. But the worst that can happen to Bill is to fall into the perjury trap: he denies the facts and Ken has hard evidence. With all this in mind we can assign numeric payoffs to all eight outcomes of the perjury game: 8
The Equilibrium Solution If Bill, Monica and Ken are rational, what should they do? First they need to make decisions in anticipation of what the others will do. And they need to be right about their expectation of the other parties behavior. In fact what the others do will determine Bill s beliefs about the decision node he reaches when time comes to discredit Monica or not. These beliefs will be based on Monica s decision to lie or not, which in turn depends on Ken s decision to give Monica immunity. Given these beliefs, Bill then decides to discredit or not by comparing his expected payoff from one course of action to the other. Let s see what GamePlan has to say: 9
GamePlan tells us that there is only one equilibrium solution to the perjury trap: If the evidence against Bill is hard, Ken gives Monica immunity, Monica tells the truth for sure and there is a substantial chance that Bill will try to discredit Monica and will fall into the perjury trap. If the evidence against Bill is soft, only Monica s decisions change since she is the only one to know for sure what the evidence is when she makes her decision. Interestingly, with no hard evidence, there is a chance that Monica will lie and that Bill will be off the hook. Both Ken and Bill take this into account when they decide what to do. 10
First, in true game theoretic logic, we look at the last move in the game, Bill s. Bill forms an opinion about the likelihood that he is at one of his decision nodes or the other on the basis of his rational expectation of what the others are doing. Bill reasons that whatever the nature of the evidence, Ken will give Monica immunity, and Monica will tell the truth with certainty if the evidence is hard. He also understands that there is only 60% chance that Monica will tell the truth if the evidence is soft. Bill knows this because he can reason things out from Ken and Monica s point of view and knows what each player s payoffs are. He therefore concludes that discredit(h) is reached with probability 0.6 while discredit(s) is reached with probability 0.4 0.6=0.24. Overall, Bill s information set and his turn in this game is reached with probability 0.6+0.24 = 0.84. Indeed, there is a 0.4 0.4 = 0.16 chance that Monica will lie when the evidence against Bill is soft, exiting the game. Bill, knowing the probability with which his two possible decision nodes are reached can now make his beliefs explicit. His beliefs are the relative probabilities with 0.6 0.24 which he is at discredit(h) and discredit(s). They are = 0. 7143 and = 0. 2857 0.84 0.84 respectively. Now he can look at his moves. GamePlan tells us that Bill does not have a clear preference for one move or the other given his beliefs. In fact if we perform the simple calculation of Bill s expected payoff from discrediting Monica using GamePlan s excruciatingly accurate belief computations, we would find that it is the same as his expected payoff from admitting to his sins: -2.28571. 2 So Bill will discredit Monica with some probability. Why 0.75? To understand this we must take Monica s priorities into account. 2 0.714286 (-4)+0.285714 2=0.714286 (-2)+ 0.285714 (-3)=-2.28571 11
If Monica looks at the soft evidence side of the story, to determine her willingness to lie she must anticipate Bill s strategy. GamePlan shows us that she must be indifferent between lying and telling the truth if the evidence against Bill is soft. Indeed she can lie or tell each with positive probability. But she can only be indifferent between these two courses of action if her expected payoff from telling the truth matches the payoff she gets from lying. So Monica must be anticipating that Bill will discredit her with probability 0.75. Now Bill s strategy is predicated on his belief that Monica will tell the truth with probability 0.6 since this determines his beliefs about his position in his information set. So Bill and Monica s expectations about what the other one is going to do justify their own choices. For an equilibrium to be reached their expectations about each other must be correct. In game theoretic lingo, Bill and Monica have rational expectations about each other s strategy choices. And Ken s too for that matter since Ken s decision to give immunity to Monica for sure enters Bill s evaluation of his beliefs which include, in turn Monica s decision to lie or not depending on evidence, which themselves are consistent with Bill s effort to discredit with 0.75 probability. It s a house of cards, carefully constructed so that each successive level of decision making is firmly supported by the decisions made below and above it. Take away one element and the structure would collapse. Decisions made would become inconsistent with each other. Each player would no longer be doing his or her best given his or her correct expectation of what the others will do. So why does Ken give Monica immunity? Since he gives her immunity with probability 1, he is sure that this is the best route for him. And this he figures by computing his expected payoff from denying immunity and comparing it to the expected 12
payoff he gets from granting immunity. OK let s do it: If Ken denies Monica immunity his expected payoff is simply 0.6-1+0.4 1=-0.2. If he grants Monica immunity and she provides hard evidence, he catches Bill red handed with probability 0.75 and Bill admits with probability 0.25. So with hard evidence, Ken s expected payoff is 0.75 4+0.25 2=3.5. If the evidence is soft, then with probability 0.4 Monica lies and Ken gets -3, with probability 0.6 0.25 he gets 3 and with probability 0.6 0.75 he gets -4. All in all, this amounts to an expected payoff of -2.55 but he only gets this with the 0.4 probability that the evidence against Bill will be soft. Giving Monica immunity yields an expected payoff for Ken of 0.6 3.5+0.4 (-2.55)=1.08. (GamePlan was right and told us exactly this at the top of the game in blue, Ken s color!). Now of course this solution is dependant on the particular payoffs we assigned to each of the players. But GamePlan is happy to let us change those, and that s what we re going to do. What Drives Bill into the Perjury Trap: Sensitivity Analysis Through sensitivity analysis we can explore the impact of our assumptions on information and preferences. The protagonists may rank outcomes differently, and may gather more or less information than we assumed at the outset. Consider information. Our perjury game assumes that Ken Starr does not know for sure whether Monica has hard evidence to give him. He makes his decision to grant immunity without knowing what the dress will yield. We could change this assumption by simply removing Ken s information set. The game now looks like this: 13
In this new version of the perjury game, only Bill is in the dark and this has interesting consequences for equilibrium behavior. The new perfect Bayesian equilibrium now has Bill own up with higher probability than he would if he believed that Ken was in the dark as well: 14
Why is Bill now ready to admit to his misdeeds with much higher probability than before? Notice that Bill s updated beliefs about whether Ken has soft or hard evidence are the same as before. But now Bill s updated belief that there is hard evidence against him relies on his expectation of Ken s behavior. If there is hard evidence, Ken gives Monica immunity for sure. With soft evidence he only gives Monica immunity with probability 0.6. By changing his behavior according to the nature of the evidence, Ken tells Bill that he should expect hard evidence with probability 0.71 when his turn comes. Now Bill exercises far more caution and discredits Monica with probability 0.28 only. 15
And this is a reflection of Ken s priorities. For Ken to be indifferent between giving Monica immunity or not when the evidence is soft, he must expect Bill to discredit Monica with probability 0.28. And Bill does so precisely because Ken s decision to give Monica immunity with probability 0.6 determines his updated beliefs when comes his turn to play. By knowing more, Ken s priorities become the drivers of Bill s behavior. When Ken didn t know about the nature of the evidence, it was Monica s priorities that determined Bill s behavior. Her tearful realization that Bill would discredit her with probability 0.875 is what leads her to lie with probability 0.4. Now that Ken will know whether the evidence is hard or soft, it is his expectation that Bill will fall into the perjury trap with probability 0.28 that determines his granting of immunity with probability 0.6 when the evidence is soft. Bill is now more likely to admit, so even if the evidence is soft, Ken s expected payoff from giving Monica immunity matches the payoff he gets from denying it. Now that Ken knows more, the chances that Bill will fall into the perjury trap are reduced. But Bill is also more likely to admit. With increased transparency Ken s darker urges incite Bill to tell the truth. What of player priorities. We can easily change player payoffs to see what happens to the solution. Of course we need to be a little careful as we do this. Remember that payoffs in game theory are typically ordinal. They reflect the player s ranking of outcomes. Meaningful sensitivity analysis in this context will therefore modify payoffs to change the ordering of player preferences but will not simply change the magnitude of payoffs without doing anything to the ordering of outcomes for the a particular player. With this in mind, let s change Monica s mindset. We initially thought that Monica 16
preferred to have Bill admit to the affair than attempt to discredit her, especially if he was to fall into the perjury trap. Instead we could think that Monica would rather be discredited if this leads Bill to finally get what he deserves: be caught lying in full view of the evidence. If the evidence is hard we will now portray Monica as preferring to see Bill deny the affair. But if the evidence is soft, Monica certainly does not want to see Bill get a chance to deny the facts getting away with all of it: she would rather lie than give him that chance. The new game could look like this: 17
Notice that Monica s change of heart might appear quite irrational. She prefers to lie when the evidence is soft, despite the benefit to Bill, than to see him get away with it by discrediting her. She feels that, then, Bill would be off the hook at her expense. You see, game theory still has something to say when emotion rather than reason drives a player s preferences over outcome. But alas, Monica s yearning for revenge will serve Bill well. By deciding that she would rather lie than give Bill a chance at salvation if the evidence is soft, Monica gives Bill the information he needs to make the best possible decision for himself. Let s look at the equilibrium solution of the game: 18
By telling the truth with certainty when the evidence is hard, and lying just as surely when the evidence is soft, Monica actually tells Bill what to expect. Just look at his updated beliefs. He knows for sure that the evidence is hard if Monica has told the truth. So he will admit and avoid the perjury trap. Technically speaking, the equilibrium is separating. Monica s behavior is entirely revealing to Bill and this ruins her plans. Naturally many other preference orderings are possible for this game. The trick, as always, is to think through the logic of the players ranking of outcomes. Internal consistency of the story you tell is key to the building of an interesting game theoretic model. Even if one of the characters has priorities that you feel do not stand to reason. 19