Games People Play. 11: Brinkmanship – The Tragic Tale of George and Saddam

Games People Play.

Brinkmanship Game You must choose a number between 1 and 100. Your number is added to that of another randomly chosen team. If the two numbers add to more than 100 both teams are out of the game. The game repeats 3 times in total. Each teams number is revealed to all the remaining teams after each round. You cannot decrease the number you choose in any round. You may not raise your number by more than 15 between any two rounds. The winners are the last team “alive” with the highest number after three rounds.

Games People Play. Brinkmanship This is an important mechanism for making incredible threats credible. A threat is a way of making a strategic move. But often you only have one large threat available. This threat is incredible because if its size, your opponent doesn’t believe you will carry it out. “Give me five dollars or I will kill myself,” may not involve a credible threat. “Give my five dollars or I will run across this road with my eyes closed,” may be credible and thus gain the five dollars. The second threat works because you are only threatening to kill yourself with a given probability (that of being hit by a vehicle). The scale of the threat is reduced and thus becomes credible.

Games People Play. Brinkmanship One of the most important elements of brinkmanship is that the process must be somewhat out of control. In the running across the road example, if the threatener could listen or look to see if a vehicle were coming then choose the probability of death by deciding when to run. But then the threat wouldn’t work because all parties would know the threatener wouldn’t run if there were a vehicle coming. They would choose the probability zero. Hence the probability of the threat actually being carried out must be somewhat outside the threateners control.

Games People Play. The Happy Tale of John and Nikita. A striking example of successful brinkmanship was the Cuban missile crisis.

Games People Play. The Happy Tale of John and Nikita. The Cuban Missile Crisis was an Example of Success of the Brinkmanship Strategy. JFK took the world to the brink of nuclear war and by doing so persuaded Khrushchev to remove missiles from Cuba. But it seems as if game theory cannot explain this. Why didn’t Khrushchev apply backwards induction, figure out the final outcome, and decide not to start the whole process? More to come….obviously.

Games People Play. The Tragic Tale of George and Saddam JFK threatened Khrushchev and forced his compliance. Few shots were fired and there was no nuclear war. He “gambled” and won. GWB played a similar strategy against Saddam Hussein “gambled” and lost. He lost because the best outcome was the compliance or removal of Saddam simply as a response to Bush’s threat. The actual outcome was the removal of Saddam but only after a costly war. Yet Bush and Kennedy played essentially the same strategy of brinkmanship. Here we might ask why Saddam didn’t employ backwards induction and quit before the war?

Games People Play. The Tragic Tale of George and Saddam We cannot say GWB in any sense played the game better or worse than JFK. Brinkmanship necessarily involves taking risks. There is a chance of success and failure. Something went wrong in the Persian gulf that didn’t in the Cuban missile crisis, but what was it?

Games People Play. A Simple Threat Model Suppose GWB and SH each have two strategies GWB - Threaten or Don’t. SH – Quit or Stay. A simple threat game then looks as follows GWB SH Threaten Don't Quit Stay GWB,SH 1, , -8 -2, 2

Games People Play. A Simple Threat Model The problem with the simple threat model is that backwards induction tells us GWB will always threaten and SH will always quit…but this didn’t happen!! Saddam didn’t quit. Why? Perhaps we have the payoffs wrong. Perhaps Saddam values defying the US and simply prefers not to quit. This suggests the problem is that Saddam’s payoffs differ from those in the game above.

Games People Play. Another Simple Threat Model. Here we see Saddam prefers to stay rather than quit. But this explanation of events doesn’t work either. GWB should apply backwards induction and choose “Don’t”. GWB SH Threaten Don't Quit Stay GWB,SH 1, , -4 -2, 2

Games People Play. Applying Brinkmanship Two elements are missing in our analysis of the tale of George and Saddam. GWB didn’t know SH’s payoffs, were they as in threat game 1 or 2? How valuable to SH it was to be seen defying the US is not easily known. SH nor GWB was sure GWB would invade. There were many imponderables including opposition to Bush’s plans in the UN by France, Germany and Russia. But how much of a check there was on Bush’s actions was uncertain to both. This leads to the following game that looks very much like the Cuban missile crisis!!

Games People Play. Telling the Tragic Tale With the two additions the game looks as right p is the probability that Saddam is tough. q is the probability that the US will carry out its threat GWB SH Threaten Don't Quit Stay GWB,SH 1, q, 2-6q - 2, 2 SH Threaten Don't Quit Stay 1, q,, 2-10q - 2, 2 GWB Nature ToughSaddam Probability p SoftSaddam Probability 1- p

Games People Play. Telling the Tragic Tale We solve the game by backwards induction. A tough Saddam will not quit since - -8 < 2 - 6q A soft Saddam will quit if 2 – 10q < -4 or q > 0.6 GWB SH Threaten Don't Quit Stay GWB,SH 1, q, 2-6q - 2, 2 SH Threaten Don't Quit Stay 1, q,, 2-10q - 2, 2 GWB Nature ToughSaddam Probability p SoftSaddam Probability 1- p

Games People Play. Telling the Tragic Tale Now consider George’s options. George doesn’t know if he’s playing against a tough or soft Saddam. If he threatens he gets p(- 2 – 8q) + (1- p)(1) If he doesn’t threaten he gets p(-2) + (1 – p)(-2) = -2 So he threatens if p(- 2 – 8q) + (1 - p)(1) > -2 or p < 3/(3 + 8q) GWB SH Threaten Don't Quit Stay GWB,SH 1, q, 2-6q - 2, 2 SH Threaten Don't Quit Stay 1, q,, 2-10q - 2, 2 GWB Nature ToughSaddam Probability p SoftSaddam Probability 1- p

Games People Play. Telling the Tragic Tale. We know A soft Saddam quits if q > 0.6 Bush threatens if p < 3/(3 + 8q) Saddam knows his type. Bush controls q. Bush may not know p.

Games People Play. Telling the Tragic Tale. Suppose Bush doesn’t know p. He experiments, first tries a q < 0.6. If this doesn’t work and doesn’t trigger conflict he experiments again. Tries a slightly higher q. If this doesn’t work, he tries again and so on. q < 0.6. If this doesn’t work and doesn’t trigger conflict he experiments again. Tries a slightly higher q. If this doesn’t work, he tries again and so on. Either he reaches q = 0.6 and a soft Saddam quits or he triggers a war. This is really no different from JFK and Khrushchev…except the outcome!!