The Critical Hit Paradox

Occasionally, while browsing 4chan's /v/ or a similar board, I'll see a thread beginning with the following game-themed math problem:

"You hit an enemy twice. At least one of the hits is a crit. Assuming a 50% crit chance, what is the probability both hits are crits?"

A crit, of course, is a "critical hit" and so a 50% crit chance is rather high, but that's not important. This is basically a variant of the second question of The Two Children Problem, also known as the Boy or Girl paradox:

"Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?"

We could also ask the same question in terms of coin flips:

"Two coins are flipped. At least one comes up heads. What is the probability that both coins come up heads?"

This last version, in my opinion, is far less contrived than one involving critical hits or child gender. Any variation of the question, however, is bound to lead to extensive debate despite its seemingly trivial nature. The question as written is ambiguous, hence the controversy surrounding the original formulation of the Boy or Girl paradox, and hence the nauseating debate that ensues whenever the game-themed version of the problem is posed to the denizens of /v/. The person starting the thread always knows this will happen, too. Much like the moving portal paradox, this problem serves little purpose other than to cause trouble and to make people angry.

But I'm more intrigued than angry. Why exactly does such a seemingly simple question cause so much confusion? Why are different people so absolutely certain of so many completely different answers? A math problem like this should have one unambiguous and provable solution, but I count three common answers on which people seem willing to bet their lives: 1/4, 1/3, and 1/2.

In an attempt to lessen the confusion, I will attempt to explain the justifications for each of these answers below. First, however, I'll summarize what we know to be incontrovertibly true about the given scenario. And, because this is a video game blog, I will be working with the "critical hit" variation of the problem, even though I think coin flips are easier to explain.

The Facts (and One Assumption)

The problem, as written, specifies a 50% critical hit chance. In other words, each attack has a 1/2 probability of producing a critical hit (or "Crit"), and a 1/2 probability of producing a non-critical hit (which I will call simply "Hit"). We are given no other combat-related probabilities, so we will have to assume that "Crit" and "Hit" are the only possible results of an attack (i.e., we cannot "miss" or anything else). This seems reasonable, as the problem is very obviously meant to be a variant of the Boy or Girl paradox, and the original question which is the subject of that paradox does not allow any genders other than boy or girl.

In summary, the possible outcomes of a single attack (with associated probabilities) are

Crit - 1/2
Hit - 1/2

Therefore, if we were simply to attack twice, we would expect the possible outcomes (with associated probabilities) to be

Crit Crit - 1/4
Crit Hit - 1/4
Hit Crit - 1/4
Hit Hit - 1/4

where we have listed Crit Hit and Hit Crit separately to preserve information about the order of the attacks (in case we need it) and, perhaps more importantly, to demonstrate more clearly that getting one Crit is twice as likely as getting two Crits. Alternatively, if we really don't care about the order in which the attacks took place, we could write the very same result as

Two Crits - 1/4
One Crit - 1/2
No Crits - 1/4

but the probability of getting two Crits remains the same, and getting one Crit is still twice as probable.

No matter how we write it, one should note that the set of outcomes described above is based solely on the stated 50% critical hit chance for every attack. Let's not forget that we also have some additional information about a particular pair of hits. According to the problem, we hit an enemy twice and at least one of those hits is a critical hit. However, we are not being told which hit is critical.

If we were told that, say, the first hit is critical, the probability of two critical hits would be exceedingly easy to find: It would simply be the probability of the second hit being critical, so 1/2 would be the answer. Equivalently, in terms of our four equally probable outcomes (see above), we can see that two of them would be ruled out, leaving only two equally probable outcomes remaining:

Crit Crit - 1/2
Crit Hit - 1/2
Hit Crit
Hit Hit

Unfortunately, this is not how the problem is phrased. We are only told that at least one hit is critical — it could be one or the other, not necessarily the first — and because of this, depending on how you understand the question, you might get an answer which is very different from 1/2.

Guaranteed Critical Hit: The "1/4" Interpretation

One fairly common interpretation of the problem (with which, for the record, I strongly disagree) is that "at least one Crit" is some kind of gameplay mechanic which is enforced during combat. In other words, in this interpretation, it is predetermined that we are guaranteed one Crit when attacking the enemy twice. Take care to note, by the way, that this is not simply a misunderstanding of what "50% Crit chance" means. Rather, it's like this: The first attack has a 50% chance to produce a Crit. If it does, the second attack then has a 50% chance to produce a Crit. However, if the first attack instead produced a regular Hit, the second attack must then produce a Crit in order to fulfill the requirement.

In other words, Hit Crit is assumed to be the only possible outcome after an initial Hit, and so it has the same probability as that initial Hit. Meanwhile, Crit Crit and Crit Hit are both possible outcomes following an initial Crit, and they share equally the probability of that initial Crit. The probabilities are, therefore,

Crit Crit - 1/4
Crit Hit - 1/4
Hit Crit - 1/2

so the probability of two Crits — and thus our answer to the question — is 1/4.

If you have Python installed, you can easily verify this result for the "guaranteed Crit" scenario using the computer simulation below. The variables first and second represent the first and second attacks, respectively; the Boolean values True and False represent Crit and Hit, respectively; the variable double keeps track of how many times we get two Crits; and the variable trial is a counter for the while loop, which iterates 100000 times. With all this in mind, the algorithm should be self-explanatory.

# Guaranteed Critical Hit (100000 trials)
import random
double = 0
trial = 0
while trial < 100000:
    first = random.choice([True,False])      # first = random
    if first:                                # if first = crit:
        second = random.choice([True,False]) #     second = random
    else:                                    # if first = no crit:
        second = True                        #     second = crit
    if first and second:                     # if two crits:
        double += 1                          #     count double
    trial += 1                               # count trial
print("Chance of two crits: "
      + str(100 * double / trial) + "%")

This simulation will indeed return a result of approximately 25%.

However, the fact that I got a computer to spit out this number does not mean it's the correct answer to the question. The algorithm used in this simulation and the logic used in the preceding discussion are based on an interpretation of the problem which, in my view, is severely flawed. There's no evidence that the statement "at least one of the hits is a crit" is any kind of gameplay rule to be enforced during combat. There's no evidence that it was predetermined before the attacks took place. All we know is that the statement turns out to be true for a particular instance of two attacks.

Furthermore, the notion of a guaranteed Crit makes no sense when we've already been told to assume that the Crit chance is 50%. This interpretation of the problem forces us to say that the Crit chance is sometimes 50% and sometimes 100%. Worse yet, while a strict reading of "50% Crit chance" would normally lead us to assume that the result of each attack (like a coin flip) is independent from the result of any other, in this solution we are forced to admit that the result of the second attack depends on the result of the first attack.

Fortunately, the next two interpretations avoid all of these problems by dropping the dubious assumption that a "guaranteed Crit" exists in gameplay.

Partial Information Provided: The "1/3" Interpretation

Another common interpretation of the problem is as follows: The two attacks take place with no restrictions other than a 50% Crit chance for each, calculated independently. The fact that we have "at least one" Crit is simply information which then happens to be true for this pair of attacks in particular.

As stated previously, when attacking twice, we should expect the possible outcomes to be

Crit Crit
Crit Hit
Hit Crit
Hit Hit

all with equal probability. Now imagine that, upon attacking twice, we don't know the actual result of each individual attack. This information has been hidden from us. All we know, after the fact, is that we happened to get at least one Crit in the process. Perhaps the game simply tells us so, or perhaps we were able to figure it out because our total damage for both attacks is known to be above some threshold. In either case, of all the possible outcomes, this only rules out Hit Hit.

In other words, of the four equally probable things that could have possibly happened, we know merely that one of them did not happen. The remaining three possible outcomes still have equal probability, and so those probabilities are

Crit Crit - 1/3
Crit Hit - 1/3
Hit Crit - 1/3
Hit Hit

Our answer to the question, therefore, is that we have a 1/3 probability of getting two Crits. It really is that simple.

As with the previous interpretation, we can use a computer simulation coded in Python to verify the result for this scenario. The variables first and second again represent the first and second attacks, respectively; the Boolean values True and False again represent Crit and Hit, respectively; and the variable double again keeps track of how many times we get two Crits. Note, however, that the variable trial does not simply count how many times we run through the while loop. It actually counts the number of runs in which we get at least one Crit. (These are the trials which are considered valid for the experiment, and the while loop runs until we have 100000 of these valid trials.) Therefore, the final result is the proportion of two-Crit trials to at-least-one-Crit trials, expressed as a percentage. In other words, the result is the percentage of at-least-one-Crit trials in which we get two Crits, and that's exactly what we want to find.

# Partial Information Provided (100000 trials)
import random
double = 0
trial = 0
while trial < 100000:
    first = random.choice([True,False])  # first = random
    second = random.choice([True,False]) # second = random
    if first or second:                  # if at least one crit:
        trial += 1                       #     count valid trial
        if first and second:             #     if two crits:
            double += 1                  #         count double
print("Chance of two crits: "
      + str(100 * double / trial) + "%")

As expected, this will give a result of approximately 33%.

Partial Result Observed: The "1/2" Interpretation

Yet another common interpretation begins much like the previous one: The two attacks take place with no restrictions other than a 50% Crit chance for each, calculated independently. Therefore, in general, we should expect four possible outcomes, each with equal probability:

Crit Crit
Crit Hit
Hit Crit
Hit Hit

This time, however, information about the result of each individual attack is not completely hidden from us. We need only check the result of an attack, and when we do this for one of the attacks, we see that it happened to be a Crit. In other words, we are not merely being told that we got at least one Crit in the process of attacking twice. We are actually discovering this fact by sampling the data, and seeing that one particular attack — either first or second — did indeed produce a Crit. Having ascertained this, we've then ruled out two of our four possible outcomes. We are left with either

Crit Crit - 1/2
Crit Hit - 1/2
Hit Crit
Hit Hit

or

Crit Crit - 1/2
Crit Hit
Hit Crit - 1/2
Hit Hit

depending on which attack we checked, and it doesn't really matter which. Once it has been established that a particular attack resulted in a Crit, the probability of getting two Crits depends entirely on the Crit chance of the other attack. The answer, therefore, should be 1/2.

Once again, we can use Python to write a computer simulation verifying this result. As in the other simulations, the variables first and second represent the first and second attacks, respectively. Each of these two variables will be assigned a Boolean value: True or False, representing Crit or Hit, respectively. The variable double keeps track of how many times we get two Crits. Finally, the variable trial is a counter for the while loop, which iterates 100000 times. For each trial, the program picks the first or second attack at random (because it should not matter which attack we sample). That attack is then set to Crit (to reflect our finding that the sampled attack is a Crit), while the other attack is randomly set to Crit or Hit (to reflect the fact that its result is unknown).

# Partial Result Observed (100000 trials)
import random
double = 0
trial = 0
while trial < 100000:
    gotCrit = random.randint(1,2)            # pick 1 or 2
    if gotCrit == 1:                         # if picked 1:
        first = True                         #     first = crit
        second = random.choice([True,False]) #     second = random
    elif gotCrit == 2:                       # if picked 2:
        second = True                        #     second = crit
        first = random.choice([True,False])  #     first = random
    if first and second:                     # if two crits:
        double += 1                          #     count double
    trial += 1                               # count trial
print("Chance of two crits: "
      + str(100 * double / trial) + "%")

The result given by the simulation will be approximately 50%, as expected.

Comparison of Solutions

Now this is a conundrum. We have three different answers — 1/4, 1/3, and 1/2 — and it's not just because I'm bad at math. The discrepancy doesn't result from any simple arithmetic blunders; in fact, the Python simulations prove that these numbers aren't just coming out of nowhere. Each of these answers follows directly from a different interpretation of the problem, and all of these interpretations seem to make some sense.

The "guaranteed Crit" interpretation, however, seems to makes the least sense. As explained before, the idea that a guaranteed Crit is predetermined (and that this requirement should be enforced during combat) is an additional assumption not stated in the problem. It's also inconsistent with the stated assumption of an independent 50% Crit chance for each swing. This alone, in my opinion, is enough to lay this dubious interpretation to rest and say definitively that 1/4 is not the correct answer. Besides, the problem discussed here is clearly intended to be a rewording of the second half of The Two Children Problem, for which 1/4 was never a candidate answer.

That original problem is actually a set of two questions. As they are stated on Wikipedia:

• Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
• Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

The answer to the first question is very obviously 1/2, because we know the older child is a girl. The probability Mr. Jones of having two girls therefore rests entirely on the probability of the younger child being a girl:

Boy Boy
Boy Girl
Girl Boy - 1/2
Girl Girl - 1/2

The answer to the second question, though it turned out to be controversial, was originally supposed to be 1/3, because we only know that at least one child is a boy. The scenario in which Mr. Smith has two boys is one of three equally probable outcomes, because all we can say is that not both of the children are girls:

Boy Boy - 1/3
Boy Girl - 1/3
Girl Boy - 1/3
Girl Girl

The competing answer to the second question is 1/2 yet again. Why two answers? It depends on how we find out that "at least one" child is a boy. The answer is 1/3 if Mr. Smith simply tells us that he has at least one boy (or, equivalently, not two girls). However, the answer is instead 1/2 if we have seen one of the children and observed that he is a boy, because then we can say that the probability of having two boys depends entirely on the gender of the other child.

If you've been paying attention, you know this is exactly what's going wrong with the critical hit problem (which, as we can see, is almost identical to the Mr. Smith question). We're getting two very different answers — 1/3 and 1/2 — depending on how we find out that we got at least one critical hit. Our intuition tells us this shouldn't matter at all, so we're tempted to say that our preferred answer (whatever it may be) is true regardless of how the information is discovered. However, we've already seen that this is not the case. Each of the two answers is totally correct if you read the problem a certain way.

Let's take another look at the "partial information provided" simulation:

# Partial Information Provided (100000 trials)
import random
double = 0
trial = 0
while trial < 100000:
    first = random.choice([True,False])  # first = random
    second = random.choice([True,False]) # second = random
    if first or second:                  # if at least one crit:
        trial += 1                       #     count valid trial
        if first and second:             #     if two crits:
            double += 1                  #         count double
print("Chance of two crits: "
      + str(100 * double / trial) + "%")

The assumption in this interpretation of the problem is that, after a particular instance of two attacks, the game has simply told us we got at least one Crit. In the simulation of this scenario, we are essentially counting double-Crit instances in a pool of 100000 attack pairs, but in building that pool, we're only considering attack pairs for which there was at least one Crit. The result of this simulation proves without a doubt that, of all attack pairs producing at least one Crit, 1/3 of those attack pairs will produce two Crits.

Now let's take another look at the "partial result observed" simulation:

# Partial Result Observed (100000 trials)
import random
double = 0
trial = 0
while trial < 100000:
    gotCrit = random.randint(1,2)            # pick 1 or 2
    if gotCrit == 1:                         # if picked 1:
        first = True                         #     first = crit
        second = random.choice([True,False]) #     second = random
    elif gotCrit == 2:                       # if picked 2:
        second = True                        #     second = crit
        first = random.choice([True,False])  #     first = random
    if first and second:                     # if two crits:
        double += 1                          #     count double
    trial += 1                               # count trial
print("Chance of two crits: "
      + str(100 * double / trial) + "%")

The assumption in this interpretation of the problem is that, after a particular instance of two attacks, we checked the result of one attack and saw that it was a Crit (and this is how we know that we got at least one Crit). In the simulation, we are again counting double-Crit instances in a pool of 100000 attack pairs, but this time the pool consists of attack pairs for which a particular attack — either first or second, at random — was determined to produce a Crit. The result of this simulation proves without a doubt that, of all attack pairs in which a chosen attack is known to produce a Crit, 1/2 of those attack pairs will produce two Crits.

It seems 1/3 and 1/2 are both correct answers. They're just answers to two slightly different questions which sound very much the same.

Bayes' Theorem

For more fun (and just as many answers), we can attempt to solve the problem using Bayes' theorem, which states

P(A|B) = P(B|A) * P(A) / P(B)

where

P(A|B) is the probability of event A, given that event B occurs
P(B|A) is the probability of event B, given that event A occurs
P(A) is the probability of event A
P(B) is the probability of event B

Once again, if we're attacking twice with a 50% Crit chance, we must consider four equally probable outcomes:

Crit Crit - 1/4
Crit Hit - 1/4
Hit Crit - 1/4
Hit Hit - 1/4

So let's say A is the event that we get two Crits. The probability of this occurring alone is

P(A) = 1/4

Now let's just say B is the event that we get at least one Crit (or, equivalently, the event that we do not get zero Crits). The probability of this occurring alone is

P(B) = 3/4

Meanwhile, the probability of getting at least one Crit, given that we get two Crits, is obviously

P(B|A) = 1

Now Bayes' theorem gives the result: The probability that we get two Crits, given that we get at least one Crit, is

P(A|B) = (1) * (1/4) / (3/4) = 1/3

On the other hand, we could instead say B is the event that a chosen attack is observed to be a Crit. The probability of this occurring alone is

P(B) = 1/2

Meanwhile, the probability that one attack is observed to be a Crit, given that we get two Crits, is obviously

P(B|A) = 1

Now Bayes' theorem gives a different result: The probability that we get two Crits, given that a particular attack is observed to be a Crit, is

P(A|B) = (1) * (1/4) / (1/2) = 1/2

Even when we turn to equations, a subtle difference can change the answer.

Conclusion

So which number — 1/3 or 1/2 — is the correct answer to our problem? I think I've proved quite enough times that 1/3 and 1/2 are each absolutely correct given the right assumption about how the stated information was obtained. However, considering the critical hit problem's roots in the Boy or Girl paradox, I think I'm going to have to choose 1/3 as the better answer. Despite the ambiguous wording, the whole point of Martin Gardner's original 1959 version of The Two Children Problem was that the two questions should have two different answers: 1/2 for the first, and 1/3 (not 1/2 again) for the second. The critical hit problem discussed here is based on the second question.

Even without author intent as backup, the decision to choose 1/3 over 1/2 is not at all indefensible. Getting an answer of 1/2 requires us to make an additional (albeit minor) assumption: that a particular attack results in a Crit. The problem, as written, only asks us to consider that we get at least one Crit. If we simply accept this stated fact, without taking it upon ourselves to imagine how that information has been revealed, we get an answer of 1/3 as explained before.

The Benefits of Cowardice

Recently, I've been playing a lot of the Gauntlet-style PC game Hammerwatch. Like The Binding of Isaac, my other recent indie game obsession, Hammerwatch came into my game collection by way of a dirt-cheap bundle whose other games I haven't touched. My digital game collection is filled with perhaps too many of those bundle B-sides — games which I only own because buying an entire set of games happened to be the cheapest way to get a single game which I actually wanted (most often thanks to Humble Bundle, Bundle Stars, and similar sites). I tell myself I'll get around to enjoying these incidental purchases eventually, but life and video games don't often leave time for each other, so it rarely happens. Sometimes takes me quite a while even to try the games I bought on purpose. For instance, I didn't actually get around to playing Hammerwatch for several weeks after the bundle went on sale.

And now, according to Steam, I've spent over 50 hours playing it. There's the first problem with Steam: It permanently records my playtime, without any option to reset the count, and displays the information publicly unless my entire profile is made private. The only way to avoid the shame of my friends knowing exactly how much of my life has been wasted is to play a game in offline mode (or run the game outside of Steam entirely if possible). The other problem with Steam is that it taunts me with achievements. Oh, sure, I can ignore achievements in a bad game. I won't play garbage just to increase the number of unlocked achievements shown on my profile. But any good game with achievements is just begging for 100% completion, and as an occasionally obsessive completionist, I often can't resist. Any set of challenges or unlockables will do the trick, in fact, but achievements — being (like the playtime counter) public and permanent — are particularly good at keeping me playing a difficult game past the point where I might otherwise have given up.

It was exactly for this reason that I found myself playing Hammerwatch's unreasonably punishing survival level, completion of which is related to two achievements (one for medium difficulty and one for hard). After the first few attempts, I began to suspect it was virtually impossible to beat, at least on my own. Hammerwatch is a multiplayer game but, having no friends currently playing the game and having no desire to play with strangers, I had been flying solo up to this point. My brother owns Hammerwatch, so I could have enlisted his help, but he hadn't played in a while and had never accumulated as much playtime as I had. He would have been rusty, at best, and might have been little more than dead weight in a game of survival with shared lives. So I continued playing survival mode on my own, determined not to let two little achievements stand in the way of total victory.

The survival level in Hammerwatch works like this: Only one extra life is given to start. Waves of increasingly numerous and increasingly powerful enemies spawn to attack the player, while the eventual boss (the Crystal Lich) sits in the center of the map, invincible but able to shoot any player who comes too close. Vendors, reached by way of a portal in a hidden room, sell upgrades and extra lives, which can only be purchased with currency obtained by inflicting damage on a few large crystals placed around the map. Meanwhile, stalactites periodically fall to the floor in random places, doing serious damage to everything in a huge area; these can kill a player instantaneously. After about 45 minutes, the regular bad guys stop spawning and the Crystal Lich comes out to fight.

My first character of choice in the game's main campaign had been the paladin (equipped with a sword which deals damage in a wide arc, a shield which blocks most projectiles coming from ahead, and some other incredibly useful abilities). However, I had heard the ranger (equipped with a long-range bow and not much else of import) was the most viable choice for beating the Crystal Lich (whose homing projectiles travel almost as far as the ranger's arrows). Unfortunately, the ranger isn't as well suited to the pre-boss fight against huge waves of enemies. The paladin would have been better for that. I could only pick one, though, and I didn't want to play 45 minutes to get to the boss only then to find myself in a virtually unwinnable fight, so I was committed to using the boss-killing ranger throughout my solo attempt.

I died. A lot. I died dozens of times without ever getting a chance to fight the Crystal Lich. After all, the ranger (who deals damage at long range but in a narrow line as opposed to the paladin's wide arc) doesn't do well when surrounded, and getting surrounded in the survival level is all but inevitable. The one obvious benefit was the ability to farm crystals somewhat effectively without stopping. The ranger can shoot a crystal while approaching and then shoot some more while departing. Even so, I could never afford enough upgrades to stay on the winning side of the arms race for very long. Eventually, I'd always start dying faster than I could farm enough crystals to replace the lives I was losing.

Then I noticed that the hidden room with the portal to the vendors, although it's a cramped dead end, is actually very safe: Few enemies spawn in range to see the player, and stalactites don't fall there (except in the case of one scripted stalactite drop which destroys the portal at the start of the boss fight). Perhaps best of all, the nova-firing trap in an adjacent room is close enough that it will fire when the player stands in the hidden room, and this periodically damages a nearby crystal for free money (as does the occasional lucky stalactite drop). This free money isn't as much as what a player can get by actively mining the other crystals, so I had no intention of hiding in the hidden room throughout the entire pre-boss battle. Still, it was the best solution for the second half of the fight, during which any attempt at mining was likely to cost me more lives than I could buy with the money I had gained.

And by retreating to the hidden room when things got too hard, taking my free money like a welfare check while waiting for the boss to appear, I finally loved long enough to fight him. At that point, it was just a matter of fighting him from a distance while finishing off any nearby enemies left over from the pre-boss phase. I won.

Thus I was left with a somewhat viable solo strategy for the survival level in Hammerwatch, using the ranger:

1. Don't destroy the nova-firing trap in the east room.
2. Farm the crystals in the north (behind the green spike trap), west (behind the red spike trap), and center (near the Crystal Lich). Each should have enough time to recharge while you're farming the other two.
• Don't bother with the crystal in the east (behind the blue spike trap); the active nova-firing trap makes it difficult to escape the room safely.
• Eventually a crystal in the south will be made available, but that little room with two small openings is a death trap. Don't go there.
3. Buy upgrades for speed, bow damage, and bow penetration. Buy an extra life when needed, but keep in mind Step 4 below.
4. When things get too difficult (and you're dying more often than you can mine enough cash for the next extra life), go to the hidden room and stand just south of the portal, ready to shoot anything that comes after you. You are not totally safe here, so don't fall asleep.
5. As you get free money from the crystal in the east (which should be taking damage from the nova-firing trap), keep on buying upgrades and/or stock up on extra lives, at your own discretion.
6. When a stalactite starts to fall above the portal, get out of the hidden room. Avoid the boss until you've cleared the nearby remaining enemies.
7. Fight the boss from a distance, coming a bit closer to shoot him and backing up to a very safe distance when he fires back. It will take a while, but if you can avoid the stalactites and any leftover enemies on the map, you'll win.

This cheesy strategy allowed me to beat the survival level on medium difficulty. Unfortunately, it wasn't so reliable for hard difficulty. After fighting and farming as much as possible without repeatedly dying, I was able to hide in the hidden room until the boss emerged, but the necessary task of clearing out remaining enemies during the boss fight became much more difficult. My damage output just wasn't sufficient to avoid being overrun once a group of enemies caught my scent, especially since the area of effect of the ranger's attack is constrained to a thin line.

Ultimately, I ended up playing with strangers to beat survival on hard mode, but even finding a suitable game wasn't easy. At any given time, I only saw one hard survival game (or none at all), and the first few games I joined were full of novices who didn't really know what they were doing. Even when I managed to join a game with more experienced players, lack of easy communication made things very difficult. By some miracle, however, I was eventually able to join a game with a couple of players (a warlock and a ranger) who were unbelievably good. Even when we were joined by a fourth player (a paladin) with no survival level experience, we weren't dragged down. I ended up getting killed before the boss was dead, but the other ranger lured the boss into a narrow hallway in which his attacks were blocked and finished him off.

So I got lucky. My advice for Hammerwatch's survival mode without friends? Try my solo strategy. If that doesn't work, I'm all out of suggestions, because you can't really count on finding a game full of expert players who are able to coordinate a victory with complete strangers. In other words: Good luck!

But, whatever. I got mine.

Love and Video Games

I'm going through a tough time in my life right now.

A year ago, I had a dead-end job. I wasn't utilizing my bachelor's degree in physics, mostly because the proper thing to do with a bachelor's degree in physics is continue to graduate school and get a master's degree or, likely, doctorate. Of course, I had been in graduate school, and dropped out for various reasons, among them severe depression and a dying interest in academia. I had decided I didn't want to teach and I didn't want to research. Getting a Ph.D. in physics not only requires both but commonly leads to one or the other. There are industry jobs as well, but I didn't have one in mind. So I had left school, and I was working, and then a year ago I decided to go back to school in another subject. For the past year, I've been studying computer science instead, with the aim of getting a master's degree in the new subject. I was confident, at first, but my present transition from undergraduate computer science coursework (in which I did extremely well) to graduate computer science coursework (which somehow isn't what I expected) is making me doubt that I can pull this off. If I can't, then I don't know what I'll do. I'm already old. My depression is coming back, which is causing problems in my relationship with my long-time girlfriend, which is causing more depression. It's not like I'm in need of psychological help. I've been through much worse and I survived. But it still sucks.

On top of all that, perhaps worst of all, is the most unbearable hardship I can imagine: I don't have time for video games anymore.

I certainly don't have time to be writing this. I mean, if I were really a good student, I would be studying or writing a paper right now. I'm not sure what prompted me to start writing this post, aside from a need to vent. I'm not even sure where I'm going with it. I guess I'll just go where my heart takes me. Outlines and structure be damned. I might not even proofread this.

Video games are a tremendous waste of time. That much is clear. All the thousands of hours I've spent playing video games over the past 20 or more years could have been spent doing much better things. I could have been studying more in high school to get straight As instead of that mediocre mix of As and Bs. I could have been working every summer instead of staying up all night every night with my eyes glued to a screen. Realistically, though, even if video games had never been invented, I suspect my eyes would have been glued to screen anyway. I'd have spent those endless hours watching TV, like a less nerdy version of a person who is exactly as lazy as I am. Are video games a more tremendous waste of time than any other non-athletic, non-educational form of recreation? If so, it's probably just because they're harder to put down.

Video games have been an important part of my life, whether I like it or not. A surprising number of my childhood memories come with a footnote of which game I had been playing that week. I once went sledding with my friends on the same day I played quite a bit of Star Fox 64, and now sledding reminds me of Star Fox 64. (Is that weird?) I talk about video games frequently with my siblings, but I often have to stop myself from doing so with my girlfriend and other people who don't care to hear how something in a game I played years ago might be relevant to the current topic of conversation where a more normal person might reference a TV show or a movie instead.

My girlfriend doesn't care about video games. She might even dislike them, perhaps even as much as I dislike her taste in music. It's not something we do together. I wish that would change, though. If she had a decently working computer, I'd try to get her to join me in some game she might enjoy, like Portal 2 or anything else with a learning curve appropriate for someone who hasn't been playing video games for at least two decades. Then again, maybe not. On rare occasions in the past, she has played a video game with me (or played alone as I watched), but I'm fairly sure now that she only forced herself to do so because she thought I'd be happy if she gave some of my hobbies a try. I'd love it if she actually enjoyed all the things I enjoy, but I wouldn't want her to pretend to like something for my sake. I haven't suggested video games as an activity for the two of us in a long time, and I guess that's why.

Is this a problem? Would I rather be dating a "gamer" instead? Absolutely not. I love my girlfriend far more than I love video games, and I think a mutual interest in something like video games is an stupid basis for a relationship. But that's just me. So you found the love of your life at a LAN party? That's amazing and I'm happy for you. For me, though, a shared interest in video games is neither sufficient nor necessary as a prerequisite for love. I'm not even convinced that it's a significant perk. I was briefly with one person who loved video games, and her interest in games didn't help the relationship work. I've been with my current girlfriend for much longer, and I've found that it's good to have something I can do without her anyway. (On a more cynical note, video games aren't going to remind me of her if we break up, so at least there's that.)

If we ever have children, she'll probably want to place strict limits on their use of the video games that will inevitably end up in our home, and I think I'd be okay with that. I could have used more discipline when I was younger. Now, like a tobacco addict who tells kids not to smoke, I know how addictive video games can be, and I don't think it makes me a hypocrite to say that kids shouldn't play them for 12 hours a day even though I used to do the same on a semi-regular basis. Moderation is key.

I don't think I could ever bring myself to ban video games from my future household entirely, and not just because I enjoy them immensely myself. No kid in the 21st century wants to have the weirdo parents who don't allow video games. Furthermore, video games are probably the cure for the inevitable phase of adolescence in which kids no longer want to hang out with their parents. As someone who always thought video games were cool, I would have been thrilled at any stage of my life if my mom or dad had wanted to play a video game with me. I don't think that ever happened, though.

The worst thing video games ever did to me was keep me from more frequently reading books for pleasure, but I can't really blame video games for simply being more fun. Now the shoe is on the other foot; my childhood is far behind me, and leading a productive life means I'm kept from the beloved pastime on which I spent far too much time in my youth. It hurts. But that's life.

The Steam Auction: Mixed Feelings

Steam's economy of trading cards and other community items got a little more convoluted a few days ago with the introduction of gems, a virtual currency which can be created by recycling unwanted items. Gems can then be used to create booster packs of cards or to bid on games in the auction that's currently acting as a weird prelude to the usual winter sale. Predictably, the whole thing was initially a disaster — an exploit caused millions of gems to flood the market and the pre-auction gem-collecting frenzy was shut down temporarily — but everything was fixed in time for the main event. The bidding began yesterday and runs until December 18th, with rounds ending every 45 minutes.

Previously, Steam trading cards were introduced in the summer of 2013. These cards are acquired primarily by logging playtime in certain games (or by spending money on optional content in free games), and they can also come from booster packs which are randomly distributed to eligible users. Once you've collected a full set of cards for a given game, those cards can then be used to craft a badge to be displayed on your profile. The cards are consumed in the process, but crafting a badge also generates some other items (which include game-themed emoticons and profile backgrounds, and sometimes coupons). If you don't want any of this stuff, trading cards (as well as the emoticons and profile backgrounds that are come from crafting badges) can be sold to other users on the Steam Community Market for money (or, perhaps more accurately, for credit to be used in the market or the Steam store). Sellers can specify prices when they list items for sale, and buyers can place buy orders at the prices they choose. Typical transactions are only a few cents per item, but foil cards and other rare items sometimes cost significantly more.

The introduction of gems, like the market itself, is good for people who just don't care about crafting badges, and would rather get rid of their cards and whatever community items they might have acquired. Now, in addition to selling their unwanted items, Steam users can recycle their unwanted items into gems. In theory, having more options is a great thing. The added confusion, though, might be a bit too much. There's already an established Steam market operating with actual currency, so introducing a secondary currency at this point is a bit weird. It's clear that gems are not meant to be a replacement for cash — you can only use gems to bid on games in this auction and to create booster packs of trading cards — but since gems can be bought and sold in sacks of 1,000 on the Steam market, they definitely count as an alternate currency that needs to be considered in certain scenarios.

Let's say I have a bunch of trading cards and I want cash. I could:
1) sell the cards on the market;
2) recycle the cards for gems and then sell the gems on the market;
3) craft the cards into badges to produce community items and then sell the items on the market; or
4) craft the cards into badges to produce community items, recycle the items for gems, and then sell the gems on the market.

Alternatively, let's say I have a bunch of trading cards and I want gems. I could:
1) recycle the cards for gems;
2) sell the cards on the market and then use the money to buy gems;
3) craft the cards into badges to produce community items and then recycle those items for gems; or
4) craft the cards into badges to produce community items, sell the items on the market, and then use the money to buy gems.

The number of gems awarded for recycling each item varies, and this value in gems is not a function of an item's (user-driven and constantly changing) market price. My trading cards for Hammerwatch are apparently worth 24 gems each, while my McPixel cards are worth only 1 gem each, but Hammerwatch cards certainly don't sell for 24 times more cash on the market than McPixel cards do. One of my foil cards, from Europa Universalis III, can be recycled for 320 gems; another of my foil cards, from Droid Assault, is only worth 80 gems, but on the cash market it's currently worth more than twice as much as the Europa Universalis III foil card.

Since different items have different gem values, and since I don't have a list of them all, I'm not sure whether it's better in general to sell items or to recycle them. For trading cards in particular, however — whether you ultimately want money or gems — I would recommend selling or crafting them, instead of simply recycling them. A typical trading card seems to be worth more in cash than in gems, given the price of a sack of gems right now. The combined market price of all the dozens of cards I'd need to recycle for 1,000 gems exceeds the current market price of a single 1,000-gem sack. Selling the cards would be better than recycling them and selling the gems produced. Similarly, selling the cards and using the cash to buy gems would be better than just recycling the cards.

Many people are, however, selling sacks of gems on the market, so clearly there are some non-card items which are worth more after recycling. In any case, whether you care about gems or not, it's generally a good idea to craft badges during events like this, since you tend to get a special trading card in addition to the regular badge and other goodies.

As mentioned above, gems can also be used to create booster packs, but this seems particularly silly to me at the current cost of gems. To create a booster pack of three Alan Wake cards, for example, I would need 750 gems, which means I would need to recycle dozens of other cards to afford it. Meanwhile, on the user-driven community market, a three-card booster pack for any given game costs about as much as three non-foil cards for that same game. If I really wanted an Alan Wake booster pack, I would be much better off just selling a few cards and then using that money to buy a booster pack from the market. Recycling dozens of cards, or even crafting full sets of cards and recycling the items that come out, seems guaranteed to be less far efficient. The only reason I'd ever spend 750 gems on an Alan Wake booster pack is to get rid of 750 gems that I can't use for anything else, since gems can only be sold in sacks of 1,000. Some other booster packs, however, cost 1,000 gems or more. There's absolutely no reason to use gems to create these booster packs as long as the market price of a sack of gems remains higher than the market price of a booster pack.

Of course, given that the currently useless Booster Pack Creator is the only way to spend gems outside of the auction, it's safe to say the market price of gems is bound to plummet after the auction ends, and this will change everything. Even if I were an expert on the ever-changing prices and exchange rates in Steam's crazy economic microcosm, it's likely that none of the advice I could provide would be worth anything by the end of the year. So you have a big pile of cards and items and you don't know what to do with them? I'm not sure that any rule of thumb exists. Just be aware of the current market price for a sack of 1,000 gems, and do some math.

So what about the auction itself? Gems are better sold than used for booster packs, at current prices, but are they better sold than used to bid on games? Not necessarily. Again, you should be aware of the current market price of gems before you place a bid. Also be aware of how much a game actually costs on the Steam store, and how much it's likely to cost during the impending sale. Some of the top bids on popular games are way higher than they should be, because some of the people placing bids on these games are out of their stupid minds. Case in point: when I checked the auction last night, the top bid for Counter-Strike: Global Offensive was more than 20,000 gems, an amount which, at the time, could have fetched considerably more than the game's $15.00 retail price if sold on the market. The Forest — a game still in Early Access (i.e., it's not even finished) — also had a top bid around 20,000 gems and a $15.00 store price.

This insanity doesn't apply to every game in the auction, though. More obscure games had top bids below 1,000 gems last night, and most of those games have store prices well above the amount that was needed to buy a sack of 1,000 gems at the time. These low-profile auctions actually seemed pretty easy to win, since there weren't many people bidding. However, for many high-profile games, you can expect at least one ridiculous person bidding a ridiculous amount of gems worth more than the game's price. In such a case, bidding is no longer worth it for anyone else except for other crazy people who place high bids just for the satisfaction of winning an auction.

I'm not really sure what's going on with these high rollers. I think you get a Steam badge for winning an auction, but if that's all they want, they could bid on more obscure games whose auctions are easier to win. Instead, they choose popular games and bid so high that they'd be better off using cash. Could it be the combination of a decent game and a badge that causes them to bid higher than a game's value? Or are they just not paying attention? Could they be blinded by the fact that real money is being substituted for artificial money for the purposes of bidding? I'm not sure.

Of course, I don't mean to imply that all these high bidders are using gems straight from the market. That is, I doubt anyone is actually bidding on a $15.00 game immediately after spending way more than $15.00 on the required gems. Many bidders probably bought gems when they were much cheaper, and recycled items to get a lot of their gems for free. However, my concern is not the amount of actual money these people are spending. It's the amount of money they could be getting instead. If, for whatever reason, you have a pile of gems which you could sell for $25.00, and you'd need all of them to bid successfully on a $15.00 game, shouldn't you just sell the gems and buy the game instead, making $10.00 in the process? (Sure, that $10.00 is just store credit, but it's better than nothing.) Better yet, couldn't you sell your gems and then wait until the sale that starts in a couple of days? That $15.00 game might be 50% off, and then you'll have a $17.50 surplus instead.

I guess I shouldn't be surprised that people are making the less rational decision. In a community of millions of people, there are bound to be at least a few hundred idiots. Honestly — forget gems — the fact that people are even willing to go to the market and spend real money on profile backgrounds and emoticons just to keep them is beyond me, but I guess I'm glad they do so. I've bought and sold some things on the Steam market (and, hilariously, made an occasional profit by buying an item and selling it later), but I didn't put any of my own money in. I started with $0.00 and a pile of cards. Now I have fewer cards and a few dollars. I'm happy with that, and I'm not about to spend those dollars on anything but a game. But maybe I just don't "get it" because I'm not an obsessive Steam badge collector.

Like the market itself, the auction is meaningless to me unless I can screw with the system to get free money. In order to win any of the most noteworthy games in this auction, I'd have to buy so many gems that the items in my inventory, whether sold or crafted or recycled, would never cover the cost. The net loss would be greater than if I just sold my cards on the market and bought the game directly from the Steam store. So it sounds like a bad idea. But, again, maybe I just don't get it.

Did I mention that the item with the highest top bid isn't even a game? It's a special profile background. You can't get it any other way, so I guess that makes it priceless, but it's not even permanent. It's only available until January 6th, after which I assume it disappears from the profiles of those who won it. The high bid, when I checked last night, was nearly 400,000 gems. At the time, these gems could have been sold on the market for well over $400.00 instead, and it would have been nearly $500.00 to actually buy them straight from the market. The high bid has been around the same amount every time I've checked since then, even though a new round starts every 45 minutes, so it's not just one person placing such a high bid. Apparently that's just how much people are willing to pay for a pretty profile background which cannot be traded, cannot be resold on the market, and disappears next month. Yeah, clearly I don't get it. I must be missing something here. Maybe the top bidders are all obscenely wealthy people who just don't care.

Anyway, if you're not obscenely rich and you're participating in the auction and you're trying to win a game, I urge you to check the price of the game on which you're bidding as well as the total amount you'd earn by selling all the gems you're planning to bid. If the latter is greater than the former, it's time to stop bidding.