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For reasons to be discussed later, limitations in their mathematical framework initially made the theory 1984 and equilibrium battle for only under special and limited conditions.
This situation has dramatically changed, in ways we will examine as we go along, over the past six decades, as the framework has been deepened and generalized. Refinements are still being made, and we will review a few outstanding problems that lie along the advancing front edge of these developments towards the end of the article.
However, since at least the late s it has been possible to say with confidence that game theory is the most important and useful tool in the analyst's kit whenever she confronts situations in which what counts as one agent's best action for her depends on expectations about what one or more other agents will do, and what counts as their best actions for them similarly depend on expectations about her.
Despite the fact that game theory has been rendered mathematically and logically systematic only sincegame-theoretic insights can be found among commentators going back to ancient times. For example, in two of Plato's texts, the Laches and the Symposium, Socrates recalls an episode from the Battle of Delium that some commentators have interpreted probably anachronistically as involving the following situation.
Consider a soldier at the front, waiting with his comrades to repulse an enemy attack. It may occur to him that if the defense is likely to be successful, then it isn't very probable that his own personal contribution will be essential. But if he stays, he runs the risk of being killed or wounded—apparently for no point.
On the other hand, if the enemy is going to win the battle, then his chances of death or injury are higher still, and now quite clearly to no point, since the line will be overwhelmed anyway. Based on this reasoning, it would appear that the soldier is better off running away regardless of who is going to win the battle.
Of course, if all of the soldiers reason this way—as they all apparently should, since they're all in identical situations—then this will certainly bring about the outcome in which the battle is lost. Of course, this point, since it has occurred to us as analysts, can occur to the soldiers too.
Does this give them a reason for staying at their posts? And the greater the soldiers' belief that the battle will be won, without the need of any particular individual's contributions, the less reason they have to stay and fight.
If each soldier anticipates this sort of reasoning on the part of the others, all will quickly reason themselves into a panic, and their horrified commander will have a rout on his hands before the enemy has fired a shot.
Long before game theory had come along to show analysts how to think about this sort of problem systematically, it had occurred to some actual military leaders and influenced their strategies.
Thus the Spanish conqueror Cortez, when landing in Mexico with a small force who had good reason to fear their capacity to repel attack from the far more numerous Aztecs, removed the risk that his troops might think their way into a retreat by burning the ships on which they had landed. With retreat having thus been rendered physically impossible, the Spanish soldiers had no better course of action but to stand and fight—and, furthermore, to fight with as much determination as they could muster.
Better still, from Cortez's point of view, his action had a discouraging effect on the motivation of the Aztecs. He took care to burn his ships very visibly, so that the Aztecs would be sure to see what he had done.
They then reasoned as follows: Any commander who could be so confident as to willfully destroy his own option to be prudent if the battle went badly for him must have good reasons for such extreme optimism.
It cannot be wise to attack an opponent who has a good reason whatever, exactly, it might be for being sure that he can't lose. The Aztecs therefore retreated into the surrounding hills, and Cortez had his victory bloodlessly. These two situations, at Delium and as manipulated by Cortez, have a common and interesting underlying logic.
Notice that the soldiers are not motivated to retreat just, or even mainly, by their rational assessment of the dangers of battle and by their self-interest. Rather, they discover a sound reason to run away by realizing that what it makes sense for them to do depends on what it will make sense for others to do, and that all of the others can notice this too.
Even a quite brave soldier may prefer to run rather than heroically, but pointlessly, die trying to stem the oncoming tide all by himself. Thus we could imagine, without contradiction, a circumstance in which an army, all of whose members are brave, flees at top speed before the enemy makes a move.
If the soldiers really are brave, then this surely isn't the outcome any of them wanted; each would have preferred that all stand and fight. What we have here, then, is a case in which the interaction of many individually rational decision-making processes—one process per soldier—produces an outcome intended by no one.
Most armies try to avoid this problem just as Cortez did. Since they can't usually make retreat physically impossible, they make it economically impossible: Then standing and fighting is each soldier's individually rational course of action after all, because the cost of running is sure to be at least as high as the cost of staying.
Another classic source that invites this sequence of reasoning is found in Shakespeare's Henry V. During the Battle of Agincourt Henry decided to slaughter his French prisoners, in full view of the enemy and to the surprise of his subordinates, who describe the action as being out of moral character.
The reasons Henry gives allude to non-strategic considerations: However, a game theorist might have furnished him with supplementary strategic and similarly prudential, though perhaps not moral justification.
His own troops observe that the prisoners have been killed, and observe that the enemy has observed this. Therefore, they know what fate will await them at the enemy's hand if they don't win.
Metaphorically, but very effectively, their boats have been burnt. The slaughter of the prisoners plausibly sent a signal to the soldiers of both sides, thereby changing their incentives in ways that favoured English prospects for victory.
These examples might seem to be relevant only for those who find themselves in sordid situations of cut-throat competition.and Equilibrium - Battle for Individual Freedom Words | 9 Pages Sarah Duhamel Mrs.
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The film follows John Preston (Bale), an enforcement officer in a future in which both feelings and artistic expression are outlawed and citizens take daily injections of drugs to. Box and Cox () developed the transformation. Estimation of any Box-Cox parameters is by maximum likelihood.
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The film follows John Preston (Bale), an enforcement officer in a future in which both feelings and artistic expression are outlawed and citizens take daily injections of drugs to suppress their emotions.