Some More on Game Theory
Some strategies that will help one draw strategic interactions between rational agents.
Not only are we in the universe, the universe is in us. — Neil deGrasse Tyson
It’s fun how we humans call ourselves the losing freaks (when we miss, say, a damn concert or score one less) and manage to devise techniques to minimise loss (I sometimes feel like, yeah, so much work, screw it, but again!). So, optimizing is more of a clever person’s approach to neglecting something of lesser importance in their life. Well, can I then think of it as a behavioural pattern?
These questions bother me, and writing them for you all has become my resolve❤Game theory is one such topic that is as entertaining as the name suggests!
Game theory is the study of mathematical models examining strategic interactions between rational (hopefully non-biased) players. These strategic interactions involve decisions made by each player, considering other players' actions and potential responses. The basic outline of Game Theory has been discussed here: https://medium.com/@phiwhyyy/concept-of-game-theory-and-saddle-points-18787f259a88 (do check it out)
Game theory aims to determine the optimal strategy for each player, considering the potential outcomes and payoffs of different actions.
Play-safe strategy
When you wish to play safe, what do you do? You prefer to look for the worst outcome and select your best option accordingly. You prioritize minimizing potential losses and focus on conservative decision-making. This strategy is often employed when the potential gains are not high enough to justify taking risks. To give some practical examples, a player with a weak hand might choose to fold rather than risk losing more money in a poker game. In a business negotiation, a company may accept a lower offer rather than risk losing the deal altogether. In essence, the play-safe strategy in game theory involves prioritizing minimizing potential losses and making conservative decisions.
Let's take an example of a game called The Prisoner’s Dilemma. In the Prisoner’s Dilemma, two individuals are arrested for a crime and are placed in separate interrogation rooms. They are given the option to either confess to the crime or remain silent. The play-safe strategy in the Prisoner’s Dilemma would involve both individuals choosing to remain silent, as it minimizes the potential loss for each player. In this case, the play-safe strategy would be for both prisoners to prioritize minimizing their potential jail time and decide to remain silent. By employing the play-safe strategy, the prisoners effectively minimise their potential losses and make a conservative decision that prioritizes their interests.
Zero-sum Strategy
A zero-sum game is a strategy where an advantage is won on one side and lost by the other side. In a zero-sum strategy, the gains and losses of one player are directly balanced by the gains and losses of the other player. This means that the total payoff of the game is constant, and any gain for one player must come at the expense of the other.
Say we have a game where two players compete for a single prize. The zero-sum strategy in this game would involve both players adopting aggressive, competitive tactics to ensure that they gain the prize at the expense of their opponent. We can formulate the problem as a zero-sum game where the players’ strategies revolve around maximizing their gains while minimizing their opponent’s gains.
Nash Equilibrium
A Nash equilibrium is a strategy profile where each player best responds to the other players’ strategies. In other words, in a Nash equilibrium, no player has an incentive to deviate from their chosen strategy because doing so would result in a worse outcome. In layman's terms,it’s more than just facts and rules; it's what the player believes- an episode of their mental state.
John Nash was a Nobel laureate who significantly contributed to game theory, including the concept of Nash equilibrium. He introduced the idea that in a game, if each player chooses their best strategy given the choices of others, then a Nash equilibrium is reached. Furthermore, Nash equilibrium is a crucial concept in game theory that focuses on the stability and balance of strategies in a game (Kuhn & Tucker, 1958).
Let’s discuss a few optimal mixed strategies in game theory. Optimal mixed strategies in game theory often involve a combination of different pure strategies to maximize the player’s expected payoff. In game theory, optimal mixed strategies are often employed when there is uncertainty about the actions or strategies of the other players. Therefore, by using a mixed strategy, a player can exploit their opponents’ unpredictability and potential weaknesses while maintaining ambiguity and flexibility in their actions.
For example, in a game of Rock-Paper-Scissors, an optimal mixed strategy would involve playing each move with equal probability. This ensures the player cannot be easily predicted by their opponent, as they are equally likely to choose any move. This strategy maximizes the player’s expected payoff by exploiting the uncertainty and unpredictability of their opponent’s moves. Another example of an optimal mixed strategy is in a game of matching pennies. An optimal mixed strategy in this game would involve randomly choosing heads or tails with equal probability. This strategic ambiguity makes it difficult for the opponent to guess and exploit a pattern in the player’s decision-making, resulting in a more balanced and unpredictable gameplay that can lead to optimal outcomes for the player.
In game theory, optimal mixed strategies are particularly relevant when players have imperfect or incomplete information about each other’s actions and intentions.
I enjoy Game Theory because it travels far beyond the books and plays with one’s intuition. Let's end this post with a problem.
Say, Phiwhyyy?!? Airlines (yes! :}) took a decision recently where they will allow the elderly (say 60+) and pregnant women to board earlier (provided they come at the allotted time), which would help their smooth boarding. This decision might initially cause a commotion but will be better for all as we know some elderly people might get disoriented in the crowd, especially if they are travelling alone. Also, pregnant women might need more time and care, especially when traveling alone. And if they decide to charge a bit extra (maybe 0.12% of the charge) for the smooth conduct of everything, it benefits both parties. What is your view of it from a game theory perspective?
The following can be considered references and inspiration for this blog:
Math Geek by Raphael Rosen
