Stakes vs Odds

Randomness in games is often interpreted as unfair by players, and in turn statisticians and game designers will often insist that the game's random number generation is perfectly fair: any player is of course going to experience streaks of bad luck, but also of good luck that they really don't appreciate enough.

I don't think players are reacting to the odds, in most cases. I think they're reacting to the stakes.

50-50 stakes (Winning equals 50 gained, losing equals 50 lost) paired with 50-50 odds (50% of the time you win, 50% of the time you lose) is exactly even and fair: you'll see streaks of wins and streaks of losses, but in the aggregate the net result of 50-50 stakes paired with 50-50 odds is that no change occurs. Someone who bets an infinite number of times with such stakes and odds will average having no change occur to their value.

I'm using this as a baseline definition of fair randomness.

People defending randomness in a game are, in every case I've seen, taking the interpretation that player complaints are focused on the odds. The usual thing is that when the game says 50-50 odds, the defenders think players believe it doesn't actually give the player a win 50% of the time, but rather 25% of the time, or some other rate below 50%.

Most games are, indeed, fair by this definition: the odds they give are the odds they mean, or something sufficiently close that it's essentially indistinguishable, regardless of whether those odds are being calculated for the player's benefit or the AI's benefit.

However, this ignores the consideration of stakes: for 50-50 odds to actually be equal and fair to the player, it needs to be tied to 50-50 stakes, and most games are not designed with equal stakes between the player and their opposition, particularly not the games that provoke player ire.

Specifically? The player stands to lose a lot more from bad luck than they stand to gain from good luck.

A reasonably straightforward example is the original X-COM, because both the player's forces and the aliens they are fighting grow stronger and wield more powerful weaponry over the course of the game, which sounds equivalent and thus fair. Indeed, all the gear the aliens get the player can acquire, and several special abilities the aliens have will also eventually become accessible to the player.

However, when a player unit dies, all the statistical improvements the soldier has acquired over the course of their career are lost as well as the armor they were wearing being destroyed, with the player having to hire a replacement soldier who starts out nearly useless, and then the player has to expend resources on replacement armor if the dead soldier was wearing armor. This is a setback that affects the player for the foreseeable future, potentially substantially if the soldier was at the upper limits of quality.

Conversely, the alien forces the player faces are completely unaffected by the loss of units, past the mission itself. Even killing rare, supposedly important aliens of the Commander rank has no affect beyond the mission it's in: the alien invasion doesn't start coming apart if the player punches enough holes in their leadership.

So if the player kills a bunch of dangerous aliens yet suffers egregious casualties in the process, the player suffers a huge setback in performance while the aliens simply ship out another batch just fine. If the player flees a mission having killed no aliens but having lost no troops, that also has no effect on the aliens: the fact that they didn't lose soldiers doesn't mean they have more available for later than they would if the player wiped them all out, for example.

In fact, in a later X-COM game -Apocalypse- killing enemies and achieving important objectives raises the player's score, and aliens respond to the player's score rising by sending out more dangerous aliens and equipping some of their less dangerous aliens with better gear!

So in Apocalypse, great success at great cost is lose-lose: not only are many of your best soldiers dead and not going to be replaced anytime soon, but now the aliens are going to be sending bigger, meaner things at you, and you're really going to wish your elite guys were still alive.

So when luck favors the player, such as by landing a very low-odds shot and killing an alien, it's a small win: that one alien isn't going to bother the player's forces for the remainder of the mission, and that's about it. Whereas when an alien gets lucky and kills one of the player's best soldiers against all odds, it carries an enormous cost. 'Lucky' events happen for both sides at the same rate, all else being equal, but the player gets almost nothing out of getting lucky and suffers tremendously from getting unlucky.

This is typical of games where random events substantially influence the player's ability to succeed. It is so much of a problem that even engineering the odds to strongly favor the player is often still unfair to the player.

The Fire Emblem series is a particularly egregious example of this: in several of the games, the listed hit chance in the game is an outright lie. Specifically, the further you move away from 50% the more rapidly you move away. A 1% chance to hit is, in actual fact, a 1 in 1000 event, rather than a 1 in 100, and by the same token a 99% chance to hit will miss once in 1000 times, not out of 100. Since the player's units are usually more accurate and better able to dodge attacks than the enemy forces, the net result is that the game is manipulating the numbers in the player's favor behind the scenes.

However, every casualty the player suffers reduces the strength of the player's army for every mission afterward, the player can only recruit a fixed number of characters so they can only sustain a certain number of casualties in total, and the player's units level-up gains are randomized not just in distribution but in total stat increase, which means one character can have a series of awful levels while another character has a series of amazing levels. This last point can be mitigated some by leveling multiple characters and then simply ceasing to use those characters who repeatedly get underwhelming level gains, but doing so means that once all your best characters are dead, the remaining characters will not only be under-leveled but weak for their level.

Enemy forces, meanwhile, will always be at a fixed level of strength and quantity that has no regard for what has happened in previous missions. The player can never get lucky and kill an important commander who thus ceases to bother them in later missions, in the same way that losing a particularly powerful character costs the player dearly.

So even though the odds are actually being actively stacked in favor of the player -doubly so, first by tending to make the player forces stronger than their enemies, and second by making the numbers favor that advantage more than they claim- the stakes are hugely different. The player gets very little out of most lucky events, but suffers major setbacks that are also quite permanent when improbable and awful events occur. It's irrelevant that the odds favor them, because the stakes are unfair.

Put in a simple way: it is unfair to place 10-90 odds with 50-50 stakes. For every fifty you win, you lose 9 fifties, putting you behind by 8 fifties overall.

But it is also unfair to place 50-50 odds with 10-90 stakes. Every ten you win is paired with losing a ninety, placing you behind by 80 overall for every single pair of gambles you make, on average.

The first of these scenarios is implicitly understood to be unfair by game designers and statisticians, but for some odd reason the second scenario is viewed as fair even though it's not. If you want to make a fair game, you need to make the odds fit the stakes, not simply provide an honest representation of the odds.


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