Two Player Games

Ever since the ancient game of Machala was invented thousands of years ago in Africa, the idea of two player games has always outstripped solo games like Solitaire.

Playing a game by yourself can be challenging, of course. The idea of beating your personal best at golf, solitaire or even a puzzle is satisfying. The increase in our confidence and self-esteem of “winning” always strikes a positive chord in our psyche.

Two player games, on the other hand have the double result of increasing our self-esteem by being challenged not only by our own skill but also against the skills of an opponent.

The online world differs greatly from the offline world when it comes to two player games, of course. The differences are not simply mechanical but include:

1. When you play an online game, it is easy to quit the game instantly and without any peer pressure. When playing online hearts with live opponents, for example, I noticed several types of people.
   1. Early Quitter. After losing a round or two, this type of person leaves the game. They either don’t realize that it is many hands that determine the games outcome or they simply don’t like losing even a single hand!
   2. Late Quitter. This person sticks around for a few hands, but once they see they are being slaughtered, they leave the game before its conclusion. They have the same qualities of the early quitter; only they stick around to insure there is no chance of recovery.
   3. Committed. Sore losers don’t congregate here. The game isn’t over until it’s over (Hearts) and you never know if an opportunity to shoot the moon will land in your lap at the end. Completing a game, even when you are bested by better players shows you stick to it until the bitter end. On the other hand, you may be learning to increase your skills or simply don’t care if the faceless person is beating you.
   4. Offline games have much more depth to the interaction than online games. One of the reasons watching Texas Hold ‘Em on the television is interesting is because we not only watch the game unfold, but we can watch the non-verbal and verbal interaction of the players. Offline two player games can be interesting to watch. Watching a single player game is definitely irritating! When I watch anyone play solitaire, I immediately want to jump in and move the cards myself!
   5. Two player games (online) allow a person to switch to a fresh new game instantly. By executing a rapid “do over” you save time, embarrassment and can increase your skills at a rapid pace. This is virtually impossible with any offline gaming experience. (Especially if an opponent is slaughtering you…they rarely want to start over!) When you rapidly start, stop and start over again during two player gaming, you often end up with a new player each time. While this won’t allow you to learn the opponent’s strategy, it can sharpen your skills and increase your understanding of the game. In the online environment, of course, you are generally anonymous so there is no shame in being an “early quitter” or even a “late quitter” as you progress.
   6. Not only can you start, quit and start over in the online two-player game world, but you can also switch the game you are playing almost instantly. By having dozens or hundreds of choices of different games, one can find out rather quickly if they like a particular game or not. If a specific game doesn’t excite you, with the click of a mouse you can be off onto a new game with new players. Only in the online world of two-player gaming can you surf, try, play and evaluate a game within minutes. While there are some offline games that include more than one system, most are single system games.
   7. The final, and most obvious differentiator between online and offline games is the investment required. Offline games (with the exception of charades, which requires more than two players anyway) require a dollar investment. Monopoly, cards, or any board game suitable for 2 or more players is a physical product that takes up physical space and a financial investment. Many online games do have a paid version, but nearly all of them start off with a completely free version you can enjoy. In fact, many of the revenue models these designers employ keep their games free forever. In this scenario, they create revenue through advertising; affiliate networks or some other system that allows complete game playing at no cost for the player. Two player games are excellent platforms for this.

Two Player Games – Playing Alone is Just Not Much Fun

If you are like me, you enjoy two player games more than solo games. While solo games are OK for starting out (or if you are in an anti-social mood), online, two-player games give you all of the choices, speed and interaction necessary to create an enjoyable, confidence-building experience. Plus, you’ll have more fun! Duh!

Two Player Games & The Prisoner’s Dilemma

When it comes to two player games, most games are mindless, action-oriented tactical missions that don’t take a ton of thought or strategy. For advanced gamers, nothing beats a game that requires strategy, psychology and multiple outcomes based on the other person’s actions. If you have the chance to play advanced games like these, nothing beats the mind-expanding game of The Prisoner’s Dilemma.

This game includes strategy, thought and a unique understanding of your opponent’s morals and decision-making skills. COOL!

The Prisoner’s Dilemma, is an aspect of game theory that shows why two individuals might not agree, even if that it is the best course of action. A classic example of the prisoner’s dilemma is as follows:

Two men are arrested, but the authorities don’t have enough evidence for a solid conviction. The authorities separate the two suspects and offer each one the same deal- if one testifies against the other, and the other stays quiet, the one who rolls over goes free and the cooperator receives the full one-year sentence. If neither of them talks, both are sentenced to a single month in jail. If both suspects ‘rats out’ the other, they both receive a three-month sentence. Each prisoner must choose to either cooperate or be quiet; the decision of each is kept a secret from the other. What should they do?

If each player is only concerned with avoiding the pokey, the situation turns into a non-zero sum game where the two players will either assist or betray the other one. During the game, weighing the pros and cons of the suspect’s dilemma is based totally on self-interest. The paradox of the challenge is the quirk that the optimal solution is for the prisoners to betray the other, despite a cooperative effort that would give them the least amount of jail time.

Regardless of what the other prisoner chooses, one prisoner will always realize a greater benefit by betraying their accomplice. In almost all solutions, betraying is more beneficial than not talking at all, therefore an objective prisoner would betray the other. Thus, the paradox on self-interest versus cooperation ensues.

In the elongated version of the game, the sequence is played over again repeatedly. Eventually, both prisoners have the choice to penalize their accomplice for his previous decision. (Payback’s a bitch!) If the number scenario’s is known, the decision players make changes. Many make the assumption that the two prisoners would betray the other over and over, regardless how many times they play. This is not always the case as some players learn the outcomes and begin to modify their decisions based on previous “wins” or “losses.”

Strategy for the classic prisoner’s dilemma

This normal Two Player games is shown below:

The table above visually represents the scenarios each prisoner gets by betraying the each other.

For example, Prisoner 1 can, with great precision, state that no matter what prisoner 2 chooses, prisoner 1 is better off rolling over on him than being quiet. The result is that for his own benefit, prisoner 1 should betray prisoner 2. Of course, if prisoner 2, acts the same way and they both act the same way, then they both get a lower reward than if both prisoners didn’t say a word.

The circular reference creates logical decisions that manifest a situation where two player games are worse off than if they each elected to reduce the sentence of their friend while penalizing himself and spending more time in jail. (How many criminals are so self-sacrificing?)

The cool aspect of the game is that neither prisoner is allowed to talk to each other. Their decisions are based on their experience, background and knowledge of each other’s personalities. If the prisoners have a high degree of trust for each other, they often choose to remain silent, lessening the penalty for both of them.

Generalized form

If you want to dissect the framework of the traditional Prisoners’ Dilemma by removing its original prisoner setting, presented as the following:

There are two player games and an impartial judge or third party. Each player holds two cards, one with the word ‘cooperate’, and the other with ‘block’. Each player gives one card to the third person, thereby getting rid of the possibility of the player’s knowing the other’s decision in advance. At the end of the turn, payments are given based on the cards played.

Based on the rules of a typical understanding of the prisoner’s dilemma, if the two players are represented by colors, red and blue, and the choices made are assigned point values it becomes clear that if the red player plays betrayal and the blue player assists the other, red gets the T prize of 5 points while blue doesn’t get payoff at all. If both cooperate they get the R payoff of 3 points each, while if they both betray they get the P payoff of 1 point. The payoffs are shown below.

Example PD payoff matrix

In simple terms, the matrix looks like this:

While on the surface, this type of theory and strategy may make your head spin, once you play it a few times, you will begin to realize how important strategy, thought and “chess-like” advanced thinking will assist you in other two player games you enjoy.

Play The Prisoner’s Dilemma with a friend tonight. Unfortunately, you already know more than he does, so you will definitely win. It will probably be more enjoyable to watch two other people play this and observe their decisions, outcomes and strategy.

In either case, you will learn a ton, refine your strategic skills and have fun playing this type of two player games!

Two Player Games – Prisoner’s Delemma

In our previous article, we discussed the strategy-building exercise of “The Prisoner’s Dilemma.” This fun and thought-provoking game develops critical thinking skills that can be transferred to any two-player game that involves some degree of strategy. (It isn’t that helpful for mindless, tactic-only games) The basic scenario is as follows: (as discussed in a previous post)

Two men are arrested, but the authorities don’t have enough evidence for a solid conviction. The authorities separate the two suspects and offer each one the same deal- if one testifies against the other, and the other stays quiet, the one who rolls over goes free and the cooperator receives the full one-year sentence. If neither of them talks, both are sentenced to a single month in jail. If both suspects ‘rats out’ the other, they both receive a three-month sentence. Each prisoner must choose to either cooperate or be quiet; the decision of each is kept a secret from the other. What should they do?

If two people play the game more than once, remember the previous moves of the other person and change their strategy accordingly, the game is called iterated prisoner’s dilemma. Sort of like “rock, paper, scissors” with a lot more thought. This is not like your typical two player games.

The iterated prisoner’s dilemma scenario expands our development of game theory especially as it relates to trust and other human behaviors. Assuming that human behavior can even be mapped or modeled, the iterated version is simply enjoyable on an entirely higher level.

When the scenario, or two player game, is executed X number times and both players know how many rounds they will be playing, then there is the possibility for each player to defect in all rounds. One might as well defect on the last turn, since the opponent will not have a chance to punish their opponent (or accomplice!). With two moderately educated players, both will most likely defect on the last turn. When X number (or finite number) of games is known, then the player might as well defect on the second-to-last turn, since the opponent will defect on the last no matter what is done, and so on.

Unlike the one-time play of the game, in the iterated prisoner’s dilemma the defection strategy is counterintuitive and fails to predict all human behaviors.

When the two player game is applied in an economic sense, however, this is the only correct answer. The rational strategy in the iterated prisoners dilemma with a fixed number of games is to cooperate against a rational opponent.

In our next article, we will discuss the four strategies used in the iterated prisoner’s dilemma:

1. Naughty or Nice
2. Roll Over or Retaliate
3. Forgive or Forget
4. Arrogant or Safe

Each of these strategies has its supporters and detractors and your personality (And, more importantly, your opponents personality) will determine not only the outcome of your games but how you arrive at your outcome…the ultimate learning experience in two player games!

Hey, who said two player games weren’t educational?

2 Player Games Strategy

Interest in the iterated prisoners dilemma (IPD) was kindled by Robert Axelrod in his book The Evolution of Cooperation (1984). This is more than a simple 2 player games. It is a study in psychology.

The author discovered that when the situation was repeated over a long period of time with more than a few players, each one with unique strategies, a strategy of self-interest or greed tended to do very poorly in the long run while more compassionate strategies did better. (Hmmm…do morals affect game theory?)

An interesting strategy is to cooperate with the first play of the game; after that, the player does what his or her opponent did on the previous move. When your opponent defects, on the next move, you cooperate. This creates a situation that allows for recovery from getting trapped in a cycle of defections. The exact probability depends on the personality and experience of the opponents. While it may appear this 2 player games is obvious like tic-tac-toe, it isn’t.

Naughty or Nice

Which makes more sense…to be naughty or nice? If you still believe in Santa, you may opt to be nice. Being nice or compassionate is sometimes referred to as an “optimistic” algorithm. Almost all of the top-scoring strategies had the characteristics of being nice; a purely selfish strategy will not “cheat” on its opponent. Remember, HIS actions also affect the overall outcome.

Roll Over or Retaliate

Being a blind optimistic player or one who simply rolls over and submits still does not guarantee the best outcome. An occasional retaliation can actually affect the opponent’s future moves. Retaliation is an overall poor choice, as “nasty” strategies will ruthlessly exploit the rollover players.

2 Player Games – Player Strategy – Forgive or Forget?

A successful 2^nd and 3^rd strategy (after the first round is completed) is to forgive. Forgetting, however, doesn’t model any learning or growth. Though players in this two player game will sometimes retaliate, they often return to a sprit of cooperating if the opponent does not continue to defect and squeal on his partner in crime. This strategy often stops long runs of revenge.

Arrogant or Safe

The quality of being safe, or not trying score more than the opponent (impossible for a ‘nice’ strategy, i.e., a ‘nice’ strategy can never score more than the opponent) has limited benefits depending on the opposing player. The optimal (points-maximizing) strategy for the one-time Prisoner’s Dilemma game is defection. However, in the iterated Prisoner’s Dilemma 2 player game an ideal strategy will always be linked to the moves of the opponents. With this personality and resultant strategy, the best tactic is to defect every time. In a population with a certain percentage of always-defectors, the great moves are scored by an individual who leans on the percentage, and on the length of the game.

Continuous iterated prisoner’s dilemma

Most work on the iterated prisoner’s dilemma has focused on players which players that cooperate or defect, because it is relatively simple to analyze. However, some researchers have looked at models of the continuous iterated prisoner’s dilemma, in which players are able to make a variable contribution to the other player. One’s perception in a continuous prisoner’s dilemma, is when a set of players starts off in a non-cooperative equilibrium, a gamer who is marginally more cooperative than non-cooperators almost no quarter from each other. The continuous prisoner’s dilemma may help explain why real-life examples of cooperation are very rare in business, nature and economics even though it would appear logical in theoretical models.

Morality

It can be assumed that morality may involve the parameter of self-interest; a player can argue that co-operating in the prisoner’s dilemma on moral principles is actually based on self-interest. It is certainly safe to give up aggressive, three-step’s ahead thinking and instead adopt a disposition of constrained maximization, according to which one resolves to cooperate in the belief that the opponent will respond with the same choice. These forms of 2 player games assumes that obvious morality-based actions have more value than being a jerk.