Monday 15 May 2023

‘Game Theory’: Strategic Thinking for Optimal Solutions

Cortes began his campaign to conquer the Aztec Empire by having all but one of his ships scuttled, which meant that he and his men would either conquer the Aztecs Empire or die trying.. Initially, the Aztecs did not see the Spanish as a threat. In fact, their ruler, Moctezuma II, sent emissaries to present gifts to these foreign strangers. 

By Keith Tidman


The Peloponnesian War, chronicled by the historian Thucydides, pitted two major powers of Ancient Greece against each other, the Athenians and the Spartans. The Battle of Delium, which took place in 424 BC, was one of the war’s decisive battles. In two of his dialogues (Laches and Symposium), Plato had Socrates, who actually fought in the war, apocryphally recalling the battle, bearing on combatants’ strategic choices.


One episode recalls a soldier on the front line, awaiting the enemy to attack, pondering his options in the context of self-interest — what works best for him. For example, if his comrades are believed to be capable of successfully repelling the attack, his own role will contribute only inconsequentially to the fight, yet he risks pointlessly being killed. If, however, the enemy is certain to win the battle, the soldier’s own death is all the more likely and senseless, given that the front line will be routed, anyway, no matter what it does.


The soldier concludes from these mental somersaults that his best option is to flee, regardless of which side wins the battle. His ‘dominant strategy’ being to stay alive and unharmed. However, based on the same line of reasoning, all the soldier’s fellow men-in-arms should decide to flee also, to avoid the inevitability of being cut down, rather than to stand their ground. Yet, if all flee, the soldiers are guaranteed to lose the battle before the sides have even engaged.


This kind of strategic analysis is sometimes called game theoryHistory provides us with many other examples of game theory applied to the real world, too. In 1591, the Spanish conqueror Cortéz landed in the Western Hemisphere, intending to march inland and vanquish the Aztec Empire. He feared, however, that his soldiers, exhausted from the ocean journey, might be reluctant to fight the Aztec warriors, who happened also to greatly outnumber his own force.


Instead of counting on the motivation of individual soldier’s courage or even group ésprit de corps, Cortéz scuttled his fleet. His strategy was to remove the risk of the ships tempting his men to retreat rather than fight — and thus, with no option, to pursue the Aztecs in a fight-or-die (vice a fight-or-flee) scenario. The calculus for each of Cortéz’s soldiers in weighing his survivalist self-interest had shifted dramatically. At the same time, in brazenly scuttling his ships in the manner of a metaphorical weapon, Cortéz wanted to dramatically demonstrate to the enemy that for reasons the latter couldn’t fathom, his outnumbered force nonetheless appeared fearlessly confident to engage in the upcoming battle.


It’s a striking historical example of one way in which game theory provides means to assess situations where parties make strategic decisions that take account of each other’s possible decisions. The parties aim to arrive at best strategies in the framework of their own interests — business, economic, political, etc. — while factoring in what they believe to be the thinking (strategising) of opposite players whose interests may align or differ or even be a blend of both.


The term, and the philosophy of game theory, is much more recent, of course, developed in the early twentieth century by the mathematician John von Neumann and the economist Oskar Morgenstern. They focused on the theory’s application to economic decision-making, with what they considered the game-like nature of the field of economics. Some ten years later, another mathematician, called John Nash, along with others expanded the discipline, to include strategic decisions applicable to a wide range of fields and scenarios, analysing how competitors with diverse interests choose to contest with one another in pursuit of optimised outcomes. 


Whereas some of the earliest cases focused on ‘zero-sum’ games involving two players whose interests sharply conflicted, later scenarios and games were far more intricate. Such as ‘variable-sum’ games, where there may be all winners or all losers, as in a labour dispute. Or ‘constant-variable’ games, like poker, characterised as pure competition, entailing total conflict. The more intricately constructed games accommodate multiple players, involve a blend of shared and divergent interests, involve successive moves, and have at least one player with more information to inform and shape his own strategic choices than the information his competitors hold in hand.


The techniques of game theory and the scenarios examined are notable for their range of applications, including business, economics, politics, law, diplomacy, sports, social sciences, and war. Some features of the competitive scenarios are challenging to probe, such as accurately discerning the intentions of rivals and trying to discriminate behavioural patterns. That being said, many features of scenarios and alternative strategies can be studied by the methods of game theory, grounded in mathematics and logic.


Among the real-world applications of the methods are planning to mitigate the effects of climate extremes; running management-labour negotiations to get to a new contract and head off costly strikes; siting a power-generating plant to reflect regional needs; anticipating the choices of voter blocs; selecting and rejecting candidates for jury duty during voir dire; engaging in a price war between catty-cornered grocery stores rather than both keeping their prices aligned and high; avoiding predictable plays in sports, to make it harder to defend against; foretelling the formation of political coalitions; and negotiating a treaty between two antagonistic, saber-rattling countries to head off runaway arms spending or outright conflict.


Perhaps more trivially, applications of game theory stretch to so-called parlour games, too, like chess, checkers, poker, and Go, which are finite in the number of players and optional plays, and in which progress is achieved via a string of alternating single moves. The contestant who presages a competitor’s optimal answer to their own move will experience more favourable outcomes than if they try to deduce that their opponent will make a particular move associated with a particular probability ranking.


Given the large diversity of ‘games’, there are necessarily multiple forms of game theory. Fundamental to each theory, however, is that features of the strategising are actively managed by the players rather than through resort to just chance, hence why game theory goes several steps farther than mere probability theory.


The classic example of a two-person, noncooperative game is the Prisoner’s Dilemma. This is how it goes. Detectives believe that their two suspects collaborated in robbing a bank, but they don’t have enough admissible evidence to prove the charges beyond a reasonable doubt. They need more on which to base their otherwise shaky case. The prisoners are kept apart, out of hearing range of each other, as interrogators try to coax each into admitting to the crime.


Each prisoner mulls their options for getting the shortest prison term. But in deciding whether to confess, they’re unaware of what their accomplice will decide to do. However, both prisoners are mindful of their options and consequences: If both own up to the robbery, both get a five-year prison term; if neither confesses, both are sentenced to a one-year term (on a lesser charge); and if one squeals on the other, that one goes free, while the prisoner who stays silent goes to prison for fifteen years. 


The issue of trust is of course central to weighing the options presented by the ‘game’. In terms of sentences, both prisoners are better off choosing to act unselfishly and remain hush, with each serving one year. But if they choose to act selfishly in expectation of outmaneuvering the unsuspecting (presumed gullible) partner — which is to say, both prisoners picture themselves going free by spilling the beans while mistakenly anticipating that the other will stay silent — the result is much worse: a five-year sentence for both.

Presaging these types of game theoretic arguments, the English philosopher Thomas Hobbes, in Leviathan (1651), described citizens believing, on general principle, they’re best off with unrestrained freedom. Though, as Hobbes theorised, they will come to realise there are occasions when their interests will be better served by cooperating. The aim being to jointly accomplish things not doable by an individual alone. However, some individuals may inconsiderately conclude their interests will be served best by reaping the benefits of collaboration — that is, soliciting help from a neighbour in the form of physical labour, equipment, and time in tilling — but later defaulting when the occasion is for such help to be reciprocated.


Resentment, distrust, and cutthroat competitiveness take hold. Faith in the integrity of neighbours in the community plummets, and the chain of sharing resources to leverage the force-multiplicity of teamwork is broken. Society is worse off — where, as Hobbes memorably put it, life then becomes all the more ‘solitary, poor, nasty, brutish and short’. Hobbes’s conclusion, to avoid what he referred to as a ‘war of all against all’, was that people therefore need a central government — operating with significant authority — holding people accountable and punishing accordingly, intended to keep citizens and their transactions on the up and up.


What’s germane about Hobbes’s example is how its core themes resonate with today’s game theory. In particular, Hobbes’s argument regarding the need for an ‘undivided’, authoritative government is in line with modern-day game theorists’ solutions to protecting people against what theorists label as ‘social dilemmas’. That is, when people cause fissures within society by dishonourably taking advantage of other citizens rather than cooperating and reciprocating assistance, where collaboration benefits the common good. To Hobbes, the strategic play is between what he refers to as the ‘tyranny’ of an authoritative government and the ‘anarchy’ of no government. He argues that tyranny is the lesser ‘evil’ of the two. 


In dicing real-world ‘games’, people have rationally intuited workable strategies, with those solutions sufficing in many everyday circumstances. What the methodologies of game theory offer are ways to formalise, validate, and optimise the outcomes of select intuitions where outcomes matter more. All the while taking into account the opponent and his anticipated strategy, and extracting the highest benefits from choices based on one’s principles and preferences.


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