RATIONAL AGENT AND THE STAG HUNT PROBLEM: IS THE RATIONAL THE OPTIMAL?

Every humanly game has its own Nash equilibrium, and this, as drawn out from game theory, is indeed an edgy discovery.

The most critical of which is that the applied subject must be a rational agent. By the book, and its own definition would suggest, the study of game theory is the study of mathematical and strategic interactions between persons with rationality [1] .

1. Rational agent.

Since for the most part, seldom are creatures-of-rationality, reflecting ourselves onto the game theory may hobble our study, then understandings on it. Our, oftentimes, rudimentary and emotional mind may easily drift us away from mathematically sound conclusions — whilst sparing rooms for falsifiable, distorted ones.

Living amidst a perplexing reality governed by the so-called morality, we have, from the outset, fenced ourselves off the optional decisions. In this manner, irrationality does not necessarily foreshadow negativity — for after all, there is still a host of motivations behind the humanly sacrificial preferences, say one’s humane benevolence and morality.

By definition, a rational agent is an agent (either a human, a pinch of AI or an organizational unit) who holds his individual priorities and models his uncertainties on the anticipated variable (or their functions’) values. Among heaps of preferences, he always goes after the ones that optimize such personal anticipations [2] .

To put into perspective, a rational agent shall never attend a casino. Having leveraged given data in calculating all the anticipated values, it seems evident that a player, however rational, could never win, for he only loses a certain amount of “capital” after each round. The optimal solution (merely economical — regardless of the entertaining factors) to this problem, as is evident, is not to play any round.

Akin to casino players who gamble their life on every call, an organization is always hung by a “decisional thread” — on an any less precarious future. That said, while the former is doomed to succumb, the latter — organizational units have every other model to assess anticipated revenues, to thereby come up with the most profitable solution. One, for example, can opt for investing revenues on stock market, opening a saving account for future expansion plans, or equal shares among their employees — the final decision of a rational organizational agent, after all, hinges upon how well he breaks down the data and draw the optimal resolution for his future.

This rationality is closely related to two-thirds of the theoretical neoclassical economics system: (1) People have rational preferences between outcomes that can be identified and associated with values; (2) Individuals maximize utility and firms maximize profits.

That said, neoclassical economics has ended up timeworn — for the initial assumptions it takes is too distant from reality, thus the most criticism is often on how theoretical it de facto is. The eboulement of such assumptions on rationality, nevertheless, did not come hand in hand with that of studies on game theory — it even bolsters the profundity of which (under the support of AI and machine learning).

This catalyzes a “rational ecosystem” wherein each individual holds his own rational preferences, which in turn must be conducive to him: Consequentialism (philosophy), Utilitarianism (philosophy), Rational choice theory (economics), Social choice theory (economics), Effective altruism (philosophy) and Game theory.

At convoluted levels on this hierarchy — philosophy for example — even seemingly moral and altruistic decisions are deduced as those driven by a host of primordial, personal advantages, instead of by the void or one’s good will. In general, whichever preferences we adopt — to either perpetuate or drop our studies, to purchase an on-sale item or not — the final decision is always the deliverable from our mental and rational struggles.

Now let us zoom out — how perplexing everything could be as rational creatures end up others’ fellow-humans and rivals and one’s action, in turn, imposes certain influences on others’ fates. To study this is the ultimate goal of game theory, as mentioned from the outset.

2. The stag hunt problem.

Game theory embodies oodles of convoluted definitions. To spare you some rooms to breathe, let us break them down one by one — article by article. Today’s, for example, is on the stag hunt problem, how we define it, along with every relevant issue.

This problem was for the first one propelled forward by Jean-Jacques Rousseau — in his discourse on inequality. A simpler variant of which is [3] :

“Two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, they must have the cooperation of their partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag, insofar as targeting at a hare may frighten stags”.

Were you a player in this game, which option would you go after to maximise the outcomes?

This game is a basic strategic game with two Nash equilibrium — risk dominance and payoff dominance.

A pure strategy is that each player makes his own decisions with a probability of either 100 or 0% (he would either do something or stay idle — a special form of mixed strategy). To adopt a mixed one, on the other hand, is to wield probability in each turn: for example, the two players then play a face-hitting game, in which the data reveals player A has a one-third probability of hitting B instead of making it plain from at the outset (either always hitting or no hitting).

The Nash equilibrium, whereas, is the balance point at which no player can switch on his strategies to optimize advantages — by playing on the other’s. A game, for example, involves two players — Potter and Hermione — with two respective strategies namely A and B. This game reaches its equilibrium once Potter is left with no alternative scheme to win advantages while Hermione is adopting B. The same is true as the latter holds no better choices while the former keeps thriving on A. The Nash equilibrium, thus, is when each player has no more-optimal strategies within a certain game.

Every humanly game has its own Nash equilibrium, and this, as drawn out from game theory, is indeed an edgy discovery.

In this equilibrium, the two dominances are risk and payoff. To better understand these terms, let us return to the stag hunt problem.

For visualizing purposes, stag and hare are hereby emblematized by numbers. Every possible circumstance can be epitomized as follows: should the two players go after stags, both end up with 3 meat units; should they both aim at hares, each only gains 1 meat unit; should one attempt hunting down hares only, he shall win back 2 meat units, whilst his counterpart wins 0 (for stags, as aforementioned, will have run away). We thereby come up with this simplified table:

_____ Stag | Hare

Stag | 3,3 | 0,2

Hare | 2,0 | 1,1

As is evident, this games is at Nash equilibrium as the players opt for stags (3,3), for one will be left with no better options [than 3 meat units], should the other go after stag (hare, in all likelihood, shall gain him as best 2 meat units). There is, nonetheless, another equilibrium as both cast the die on hare (1,1): if A has opted for hares, B holds no option other than also hunting down hares (as aiming at stag would otherwise gain him nothing).

(This spells out the difference between this problem and Prisoner Dilemma, in which the only Nash equilibrium is to play Judas).

The aforementioned payoff dominance in this game is (3,3), while (1,1) displays the risk dominance. Whilst zeroing in stags straightens out one’s potential payout, a hare-hare option curtails his risks. It then seems an arduous task to make up his mind on any decision.

The bona fide anecdote of stag hunt is rather intriguing, and no less familiar with what actually takes place in reality:

A group of hunters have gone after a giant stag for a while, and eventually ferreted out its traces. Should they all cooperate, triumph for humans is doomed and everyone will have their own shares. Otherwise, it would run away and they all would die of starvation.

Thereafter, they patiently ambush the stag on this track. One hour has passed and the beast has yet to turn up, then two, three, and four hours of waiting in desperation. They now have been ambushing it for one day. They all know that it does not necessarily go on this track everyday, yet the stag will inevitably pop by one day. Out of the blue, one hare pops in.

Should one of them attack it, he will earn food for himself, whereas the pre-installed stag trap will be nullified and others, in turn, will die out. There is no safeguard that the latter would turn up “in time”, whilst the tempting hare is available. The risk, then, lies not only in the woefully low possibility of the stag’s debut but also that of the hare as a bait. A moment of impatience from any hunter would drive every other starved to death. In the end, while the stage’s turning up remains largely unlikely, what seems prone is that anyone, at any time, can lose their temper and fall for the hare — which is straightforward enough, since every hunter would bear with that very same fear. Were you a member of which, how would you move?

There is also a host of kindred assumptions: given two roommates, should both stay uncluttered, the room, in turn, will stay in good shape. Should one do, while the other do not, the former’s effort will be in vain. This also applies to when it comes to the household chores.

Since the game is pre-designed with two Nash equilibria, neither of which appears the “smarter” move. The lackadaisical, to put into perspective, have every reason to delay cleaning rooms, as well as their cluttered lifestyle. And since this is a game with two balance points, there are always those going after the optimal advantages, and their counterparts — the least risks. Under game theory’s perspective, it turns out any less comprehensible why the housework dilemma between roommates has been so ubiquitous.

A such game is pre-installed with two winning scenarios, and so, both would happen.

While one’s denial of houseworks and drags himself, as well as others, into horrible situations, oftentimes only gets on our nerves — we reckon they can always opt for cooperation to better theirs and others’ lives — how would we act upon this in games with much higher risks, for example, a hunter confronted by starvation, or a nation putting itself on line upon failures to keep pace with its neighbors in the weapon race? Are the motives towards collaboration (while taking onerous risks), indeed more desirable than walking out on the so-called morality or oburdening the budget?

Even when we are de facto “creatures of rationality”, making any decision has never been an undemanding task.

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References:

[1] “game theory | Definition, Facts, & Examples | Britannica,” Encyclopædia Britannica. 2020, Accessed: Dec. 15, 2020. [Online]. Available: https://www.britannica.com/science/game-theory.

[2] Wikipedia Contributors, “Rational agent,” Wikipedia, Dec. 15, 2019. https://en.wikipedia.org/wiki/Rational_agent (accessed Dec. 15, 2020).

[3] Wikipedia Contributors, “Stag hunt,” Wikipedia, Nov. 06, 2020. https://en.wikipedia.org/wiki/Stag_hunt (accessed Dec. 15, 2020).‌

Further Reading:

- Rational choice theory: https://en.wikipedia.org/wiki/Rational_choice_theory

- Strategy (game theory): https://en.wikipedia.org/wiki/Strategy_(game_theory)...

- Nash equilibrium: https://en.wikipedia.org/wiki/Nash_equilibrium