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Analysis of the Ambidex Game

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Ally-or-Betray2

This article looks at the Ambidex Game, as features in Virtue's Last Reward, from the point of view of purely rational play. You probably know that the Ambidex Game is based on a classic game theory problem known as the Prisoner's Dilemma, which demonstrates a case in which rational reasoning fails. However, in the Ambidex Game, rational reasoning can actually work better than it does in the Prisoner's Dilemma.

Disclaimer: This article supposes purely rational players, whereas in the game Virtue's Last Reward, the players do not play purely rationally. The Ambidex Game is largely used as a storytelling device to create suitable endings and plot twists.

The Prisoner's DilemmaEdit

The original Prisoner's Dilemma was described in terms of two prisoners, each in their own cell, and unable to communicate with each other. A police officer visits each of the prisoners and tells them that they have the opportunity to betray the other prisoner by informing on them. This will affect their prison sentence as follows:

  • If both prisoners inform on each other, both will go to prison for 8 years for their crimes.
  • If neither prisoner informs, they will both go to prison for 2 years on a lesser charge.
  • If one prisoner informs and the other does not, the informant will be set free for cooperation, while the other prisoner will go to prison for 10 years.

Unlike the Ambidex Game, the original Prisoner's Dilemma has only one round. The prisoners either betray each other or not, and once they have decided, the game is done.

The points table for the original Prisoner's Dilemma looks like this. The reward before the / is what you get, and the reward after the / is what your opponent gets:

Opponent Allies

Opponent Betrays
You Ally -1 / -1 -3 / 0
You Betray 0 / -3 -2 / -2

Note that all the numbers are negative, since they represent a number of years in prison, which is a negative thing from the point of view of the players.

The main interest in the original Prisoner's Dilemma is that there is no rational approach to choosing an answer. Looking at the points table, it seems that betrayal is the best option, since at worst you will go to prison for 3 years, and at best you might go free. However, the problem is that your opponent will probably think exactly the same way, and thus you will certainly go down for 3 years - whereas you could have had only 1 years by allying. However, if you then reason that your opponent will realise this also, and will therefore ally, then betraying becomes the best option again because it will let you go free - and your opponent realises that too, and so on ad nauseam. The problem becomes insolvable.

The four different circumstances that can arise in the game are given formal names to identify them. The case where both players ally (what Zero calls the "Best Pals" outcome), is called Mutual Alliance. The case where both players betray is called Mutual Betrayal. The case where you betray and the opponent allies is called Temptation: it offers a higher reward and is part of the reason to betray. The case where you ally and the opponent betrays is called Sucker, since it means you lose out.

(Mathematical explanations usually use the terms Cooperation and Defection instead of Allegience and Betrayal, but this article will stick with the later two terms because they match the game.)

The Iterated n-Prisoners' DilemmaEdit

The variant of the Prisoner's Dilemma on which the Ambidex Game is based is formally called the iterated n-prisoners' dilemma. N-prisoners indicates that there are more than two prisoners; iterated indicates that the game is played more than once. This complicates the game in a number of ways: in particular, it's possible to predict a person's behaviour based on their previous actions, and to take revenge on a player who has betrayed you in the past.

Although there isn't a good solution to this, it is used as a simulation base for testing algorithms and behaviour related to trust between people or groups of people.

The Ambidex GameEdit

The Ambidex game uses the following reward table:

Opponent Allies

Opponent Betrays
You Ally +2 / +2 -2 / +3
You Betray +3 / -2 0 / 0

This follows the standard pattern of the Prisoner's Dilemma. The Temptation outcome (+3) is better than the Mutual Alliance (+2), which is better than the Mutual Betrayal (0), and the Sucker (-2), which creates the conditions for the Prisoner's Dilemma.

However, not all players always experience the chart in these terms. This is because the Ambidex Game adds three Boundary Conditions which can disrupt the chart. The boundary conditions are:

  • Escape, if you reach 9 points or above. This is better than any other gain or loss of points.
  • Trapped, if someone else reaches 9 points or above and you don't. They will likely escape and the game will end with you trapped in the facility. This is worse than any other gain or loss of points.
  • Death, if you go to 0 points or below. This is also worse than any other gain or loss of points.

The Boundary Conditions cause several changes to the behaviour of players in the Ambidex Game. For example, suppose that you are on a score of 1 or 2 points. The chart for the game looks like this:

Opponent Allies

Opponent Betrays
You Ally +2 / +2 Death / +3
You Betray +3 / -2 0 / 0

Since the Sucker reward is now Death, you are almost certain to betray in this position - as long as you betray, you won't die, although you might not get any points. This gives the interesting consequence that no rational player in the Ambidex game ever dies because they would need to Ally in this position and they will not do so. This is similar in its moral consequences to the Nonary Game in 999 and before: with the implication being that the creator of the game does not actually intend any of the players to die.

Now suppose that you are on a middling score - 5 or lower, but your opponent is on a score of 7 or 8 points. The chart you face looks like this:

Opponent Allies

Opponent Betrays
You Ally Trapped / Escape Trapped / Escape
You Betray +3 / -2 0

If your opponent gains anything, they will escape and you'll be left trapped. So again, in this case, you have no rational choice but to Betray, so that your opponent can only lose. The same happens if your opponent is on 6 points:

Opponent Allies

Opponent Betrays
You Ally +2 / +2 Trapped /

Escape

You Betray +3 / -2 0

If your opponent betrays you, they will escape. This means that they're extremely likely to betray, and if they do and you ally, you'll be trapped, so again you're sure to betray.

This gives an interesting situation in that once someone is on 7+ points, and if all players are rational, there is only one circumstance in which they can escape. In almost every case, their opponent will choose to betray them to prevent themselves getting trapped, and thus the high-scoring player cannot gain anything and cannot reach the 9 points needed to escape.

The one situation in which such a player can escape is if your opponent has 7-8 points and so do you. The chart becomes:

Opponent Allies

Opponent Betrays
You Ally Escape / Escape Trapped / Escape
You Betray Escape / Trapped 0

The significant change in this chart is that there is now no Temptation to Betray. Having 10 points is no better than having 9 if escape is your goal. Although betraying removes any risk of getting Trapped, if the rational response to this situation is to Betray then both players will always do it they will never escape. Although knowing that the opponent will Ally would allow you to betray and prevent them escaping, preventing an opponent from escaping seems to have relatively low value compared to the benefit of escaping yourself. So in this situation, the two players will mostly likely Ally.

In other words, the only way to escape is to have 7-8 points, and play against someone else who also has 7-8 points, and who trusts you.

Having 6 points, on the other hand, is much nastier. Suppose that your opponent has 6 points, and you have 7-8.

Opponent Allies

Opponent Betrays
You Ally Escape / Trapped Trapped / Escape
You Betray Escape / Trapped 0

Your opponent will surely betray you; since, If they ally, they will certainly end up trapped. Since you can predict you will be betrayed, you will also betray to avoid getting trapped and the result will be 0.

This has an interesting consequence: if you're on 6 points, everyone will always betray you. And since you can only gain points if your opponent allies, the effect is that if you're on 6 points you're effectively stuck! You will be mutually betrayed every time. In fact, on 6 points, the only real option you have is to Ally and accept the loss of 2 points.

So, now let's look at the game at the starting condition. You and your opponent, whoever they are, are both on 3 points. Let's look at the number of points you'll be on after each outcome, and the situation it leaves you in:

Opponent Allies

Opponent Betrays
You Ally

5 / 5 (no rules observed)

1 / 6

(you and your opponent are both stuck*)

You Betray

6 / 1

(you and your opponent are both stuck*)

3 / 3

(this same situation repeats again)

  • As stated above, a player on 6 is stuck because they will always be betrayed as opponents wish to avoid being trapped. A player on 1 is stuck because they must always betray to avoid death. Since the opponents will be able to predict this betrayal, the opponent will betray too and the resulting score will be 0, leaving the situation unchanged.

So the only desirable outcome comes from mutual alliance.

If you ane your opponent are on 5 points each, as they will be after the previous play:

Opponent Allies

Opponent Betrays
You Ally

7 / 7

(ally with each other again and you can both escape)

3 / 8

(your opponent can escape by allying with someone)

You Betray

8 / 3

(you can escape by allying with someone)

5

(this same situation repeats again)

Whether or not this is a true prisoner's dilemma situation depends on your relationship with players other than your opponent. Being able to escape by allying with your opponent again is a desirable situation because that is certain to eventually happen. Being able to escape only by allying with a random other player - who has seen you achieve your position be betraying someone - is of very variable value.

Logically, however, the only choice that ever truly makes sense is to ally.  The reasons for this are thus, assuming the game has just started and all scores are still equal; 1. Betrayal is practically guaranteed to extend your game and drastically lower your chances at winning.  If you were up against someone who betrayed in the previous round, you understandably wouldn't trust them.  This is a poor outcome for everyone.  But more importantly 2.  If you choose betray with the intent to gain points, then you already expect your opponent to Ally.  This, combined with the previous fact, means that if everyone were making truly logical choices, even in the self-interest of escaping as quickly as possible, the scoreboard would show nothing but Ally for each round.

Determined plays by scoreEdit

Their Score >

Your score

1 2 3 4 5 6 7 8 Conflicted
1 Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death)
2 Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death) Betray (avoid death)
3 Sacrificial Ally (save other?) Sacrificial Ally (save other?) Ally (avoid 6 points) Ally (avoid 6 points) Ally (avoid 6 points) Betray (avoid entrapment) Betray (avoid entrapment) Betray (avoid entrapment)
4 Sacrificial Ally (unstick) Sacrificial Ally (unstick) Betray (avoid 6 points) Betray (avoid 6 points) Betray (avoid 6 points) Betray (avoid entrapment) Betray (avoid entrapment) Betray (avoid entrapment)
5 Betray (will be betrayed) Betray (will be betrayed) Ally (prep a partner) or Betray (get a start)? Ally (prep a partner) or Betray (get a start)? Ally (prep a partner) or Betray (get a start)? Betray (avoid entrapment) Betray (avoid entrapment) Betray (avoid entrapment)
6 Sacrificial Ally (unstick) Sacrificial Ally (unstick) Sacrificial Ally (unstick) Sacrificial Ally (unstick) Sacrificial Ally (unstick) Betray (avoid entrapment) Betray (avoid entrapment) Betray (avoid entrapment)
7 Betray (will be betrayed) Betray (will be betrayed) Betray (will be betrayed) Betray (will be betrayed) Betray (will be betrayed) Betray (will be betrayed) Ally (no benefit from betrayal) Ally (no benefit from betrayal)
8 Betray (will be betrayed) Betray (will be betrayed) Betray (will be betrayed) Betray (will be betrayed) Betray (will be betrayed) Betray (will be betrayed) Ally (no benefit from betrayal) Ally (no benefit from betrayal)

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