Elitist Jerks Tol Barad, the Prisoner's Dilemma, and Cooperation

# Tol Barad, the Prisoner's Dilemma, and Cooperation

Posted 12/28/10 at 4:09 PM by Hamlet
Introduction

In response to the fact that Tol Barad has turned out to be clearly imbalanced in favor of the defender, Blizzard implemented the following change: winning as the attacker rewards 1800 Honor Points (and winning as the defender still rewards 180). For purposes of this analysis, the only thing you need to know about Tol Barad is that the winner of any match will be the defender in the next match.

I'm going to do some brief game-theoretic analysis to show that the defending side now has a significant incentive to throw the match. When I first read this change, I thought "this creates a perfect Prisoner's dilemma - Wikipedia, the free encyclopedia ." It turns out that that's not the case--as I'll show here, it creates something that incentivizes losing even more than the canonical Prisoner's Dilemma does.

The Prisoner's Dilemma

If you're not at all familiar with the Prisoner's Dilemma, skim the wikipedia article linked in the last paragraph. It's a fundamental game theory problem which has been the subject of extensive literature over the past few decades. Briefly, it concerns a game played between two players, each of whom has two choices: "cooperate" with the other player, or "defect" from him. It is a non-zero sum game, meaning that one player's win is not necessarily the other's loss--the outcome can be simultaneously good or bad for both. In the classic Dilemma, the best total result for both players is if they both cooperate. However, there is a powerful incentive to defect, since doing so can let you claim a large prize at the expense of the other player. But if both players defect, the result is worse than if they both cooperate. For more details, see the wikipedia article.

I'm going to lay out an analysis of Tol Barad in the following way: there are two sides (Horde and Alliance) who are going to play iterated rounds of the Tol Barad game. Each side has two possible strategies:
Cooperate: Play normally when attacking, intentionally lose when defending.
Defect: Play normally (trying to win) always.

What we need to is compute the expected outcome for both sides under all combinations of cooperating and defecting. As one intermediate step, we need to know the probability that the attacker will win if both sides are trying. Because Blizzard has made the reward for the attacker 10 times are great as that for the defender, they've implicitly assumed that the attacker as a 1/11 chance of winning, so I'm going to go with that.

1) Both sides cooperate: the attacker wins every game. Each side gets 1800 points in alternate rounds, for an average payout of 900 points per game.

2) Both sides defect: the game is played normally. On average, Tol Barad changes hands every 11 games. Therefore, in each set of 22 games, each side can expect to win 1 game as the attacker and 10 as the defender. Each side gets an average payout of 3600 points over 22 games, or 163.6 points per game.

3) One player cooperates and the other defects. Remember this is the case in which one side voluntarily throws all of their defender-side games, seemingly for nothing in return. So whenever they have Tol Barad, they give it away next round, and take an average of 11 rounds to get it bad. In each set of 12 games, the cooperator expects to win 1 game attacking for 1800 points. The defector wins 1 game attacking and 10 games defending, for 3600 points. The average payout is 150 points per game for the cooperator and 300 for the defector.

Summarizing the results in a table:

 P1\P2 Cooperate Defect Cooperate 900,900 150,300 Defect 300,150 163.6 163.6

(Note that this is not a true Prisoner's Dilemma, since the 900 is greater than the 300. This makes cooperating look overly attractive, which is part of the point of this post).

So the first result is obvious--the two players working together have a strictly dominant strategy in which both cooperate. If the Alliance and Horde on any server can come to an agreement, there is no question that they should agree to perpetually have the defender throw their games.

More likely though, no actual agreement can be reached. The more interesting question is how to behave when you don't know what your opponent will do. If you assume they will defect (because they have not read this post), your are better off defecting (getting 163.6 points per game over 150). However, it is rather notable that the difference is very small--even you expect to get no consideration in return, there is almost no penalty in throwing all your games while defending! And if you think there's even a small chance that your opponent might cooperate (for example, if you spread this information through your server community, so at least the occasional enemy team might cooperate), the balance rapidly tips in favor of cooperating.

Conclusion

So your goal, really, is to find some way to plant a "seed" of cooperation on your server. If you can get over a threshold where even a small portion of Tol Barad groups elect to give up when on defense, the system will snowball towards complete cooperation: that promised land where everyone can get 900 points per game.

Obviously there are practical difficulties in getting something like this to happen in the live community. A substantial fraction of the server community will have to have read this post or something like to understand why they would intentionally lose Tol Barad. But I'd be highly interested if any server (maybe a small one) could ever reach the point where it becomes common practice for Horde and Alliance to alternate TB victories to secure the maximum rewards.
Posted in Math

 Very good analysis! I've dealt with Prisoners Dilemma during graduate study myself in a more qualitative than quantitative capacity. In the context that you establish you are spot on and I agree with your conclusions. As you have already thought out the theory would be difficult to deploy due to communication issues and the need for both factions to at least reach a tacit agreement to engage in such a back and forth. There are two additional wrinkles I'd add. The first being that not necessarily all TB participants value maximizing their Honor gains. Some players maybe interested in completing achievements or winning zone for access to the raid and may work against the tacit back and forth exchange. Even if it's just one player they alone maybe sufficient to tip the balance of the battle and possibly break the carefully established back and forth agreement. Second, the participants in the model you established are not fixed. Each iteration of the game will effectively have a unique composition of players so there maybe "structural" problems in the combat (e.g. too many healers on the attacking side, etc). Additionally, this aggravates the problem I point out in the previous paragraph. New participants in every iteration maybe ignorant that they must "throw" a particular match in order to maximize long term honor gains. Regardless, the fix that Blizzard has done replaces one problem with another. The incentive to constantly defend is now replaced with an incentive to constantly "conquer" which is only possible if you have lost the previous round. A balance issue in victory condition mechanics may now turn into something else. Such a tit-for-tat agreement on a server defeats the purpose of the zone thus turning it into more of a commodity than anything else. You are absolutely correct in your analysis that given the way Blizzard has now structured TB players would do themselves a favor to as quickly as possible setup such arrangements on their respective servers to maximize their honor gains; clearly though that's not what Blizzard was trying to aim for. Posted 12/29/10 at 3:37 AM by emptyrepublic
 This is a really excellent analysis of the current Tol Barad setup, with regards to maximizing potential honor gain. As Cataclysm content matures, I suspect that the focus on honor will increase, but currently there are rewards other than honor that people are concerned with which will also be influencing their behavior in TB. As Emptyrepublic mentioned, alternate goals/concerns make the traditional Prisoner's Dilemma more complex. Aside from achievements and raid access, currently players are also concerned with reputation and commendation gains, primarily for access to epic trinkets and mounts. On top of the quick rep/token gain for a win, the potential for additional daily quests for bonus rep/tokens will provide some players motivation to keep winning until they have access to each possible daily. With different play schedules (Emptyrepublic's variable participants) and randomly available dailies, on a high pop server some players will be pushing for victory almost constantly, since TB dailies and wins are the only possible source for TB rep and tokens, while honor farming can be accomplished in other ways. I could be wrong about the popularity of the TB commendation rewards, or the dedication and number of the players seeking these rewards. Since we can't survey every player who runs TB, I suppose we will just have to look at long-term trends, and try to figure out why any shifts in the win/loss ratio may occur. Posted 12/29/10 at 8:29 AM by Calinette
 I want to echo the comments above. While your analysis is true if the goal is to maximize Honor earned form TB, I doubt Honor is a primary motivation for most of the people running TB. To use myself as an example, I'm doing TB as often as I can for PvE rewards. The Hit trinket is quite good for a starting trinket, and the Tanking trinket has quite a bit of situational utility even for a DPSer. The resistance could be helpful in some fights, and my guild is encouraging all raiders to work towards one. As a result I need lots of rep and lots of tokens. The first time I go to TB to do dailies I have access to 11 or 12 dailies that provide rep and tokens. The next time Horde wins I potentially gain 3 more dailies, and if I participate in the win, I gain additional tokens. After that win there is the potential to gain 3 more dailies for a total of 17 or 18. Sure, I could gain assess to these additional daily quests with cooperation, but I am a father and a working professional. I don't have time trade it back and forth. I need to try and pick all of them up in a period of about 5 hours. Therefore, I need to win TB each time it is contested, if I want to maximize Rep and Tokens gained. Real PvPers on the other hand will probably focus on more traditional forms of PvP. The 1800 Honor gained is unlikely to attract a lot of them since TB is still very hard to attack if the other faction has a reason to defend. The probably will be able to get an similar amount of Honor with BGs. Posted 12/29/10 at 11:54 AM by Cdin Updated 12/29/10 at 11:59 AM by Cdin
 One of the conditions of the classic prisoners' dilemma is that they can't communicate to work out an optimal strategy. While on the spot communication between opposing factions is difficult, it's really not hard to communicate cross-faction outside of the game, or even within it by using alts. Because of that, we drift even further from the game theory problem. Further, most servers have a small number of recognized, respected, and visible guilds. Members of those guilds are among the most likely to be in contact with members of similar guilds from the opposite faction, meaning they can work out such agreements easily, and use their position within the server community to spread the word and influence the behavior of others. On my server (Dalaran US), if Ascend leads a handful of other Alliance guilds into an agreement with Doom (top Horde guild) and Legitimate (2nd Horde), most of the smaller guilds and unguilded players will follow. But even if they don't follow it would only take a portion of the defense team working to throw the battle. In a raid of 40, if you have 15 from the 'agreeing' guilds (usually the best geared and often quite competent) sitting out the fight, the remaining 25 aren't going to have much of a chance of defending, especially since the attackers will include those 15 well geared and competent players. But, at least on my server, there didn't even have to be that much work put into it. TB started swapping back and forth almost immediately, with no coordinated plan to do so, as far as I could tell. People just grasped that they were better off losing, when it was explained to them. Posted 12/29/10 at 1:23 PM by Kurgosh
 There are times where I - as a defender - have no incentive to lose regardless of the state of the next game, or future rewards. If I stop playing at 10pm every night, and there is a TB starting at 9pm - it's best for me to (try to) win that match. I won't be playing the next match so don't care if my side gets 180 or 1800 points from it. I'd rather take my 180 and go to bed, than take zero so my team can get 1800 next time. Same thing if I have a raid in 1 hour, etc, etc. The above is assuming that enough players will be online and queueing for TB within the time frame for each side to win AND lose twice each. (that's 8-10 hours). In reality very few individual players will be able or willing to queue up for 4 consecutive TB battles. Even assuming one plays 4 matches in a day, once they've successfully attacked twice for the day, why would they stay to lose a 4th match, just so their team can win the 5th, which they will not be participating in? Each player needs to be willing to participate in the current game, plus the next two games for the "trade" system to work. Any player is only willing to stay for 1 or 2 games, has absolutely zero incentive to lose. The prisoners dilema assumes each player will have the same amount of turns. In TB this is practically never the case. The penalty for throwing a game can mean zero points for you at all in a given day - if your'e playing 1 or 2 games (1/11 chance to take it back next time. While winning gaurantees at least some points in a 2 game span. The odds of getting players willing to throw the game are much smaller than those trying to win - regardless of attacking or defending, and regardless of the maximum possible points able to be gained. Posted 12/29/10 at 1:43 PM by Andenthal
 Except you don't need the entire team to throw it. If even 25% of defenders don't defend, you should be able to take it. We have been win trading everytime I've played anyway for at least a week on Kel'thuzad. No formal agreement, it just seemed patently obvious. There's no point to defending when if you lose, you get 10 times the honor next time. Sure not everyone will be there next time, but enough people will be. I went to my old server Nordrassil, leveled my pally, qued TB and was SHOCKED that alliance were actually defending. The successfully defended too, they went all the way. Yup they got 180 honor in over 30 mintues. Gratz them, if they'd lost they could have got 1800 in 2 minutes next time. Posted 01/02/11 at 6:32 AM by Greentouch
 Recently the value was changed to 360 honor for a win. I assume this idea spread a little too fast . Posted 01/05/11 at 12:59 AM by frmorrison