A version of this article first appeared in the May 2010 issue of Game Developer magazine.

In 2004, the Red Sox were down 4-3 in game 4 of the American League Championship Series in the bottom of the ninth inning.  Their hated rivals, the Yankees, had won the previous 3 games and were 3 outs away from going to the World Series.  But in one of the most thrilling championship series of all of sports, the Red Sox managed to tie the game in the ninth inning, win the game in the 12th, and proceed to win the ALCS and then the World Series.  Seriously, what are the odds?

Thanks to a bunch of fans, we know exactly what the odds are.  Baseball as a sport lends itself to statistical analysis – a hobby advocates call Sabermetrics.  Unlike other team sports, baseball has a relatively low number of variables – it really comes down to the duel between the pitcher and batter – so it’s easy to quantify the value of any given player in any given situation.  There are few parts of the game that have not undergone statistical analysis.

So when we ask, “what are the odds the Red Sox would win game 4?”, we know exactly: in the history of all baseball games that have ever been played, a team that has been down one run with no outs at the bottom of the ninth inning has won 23% of the time.  When Kevin Millar got his walk, that percentage jumped up to 37%.  One stolen base later — making the situation a runner on second with no outs – and  the odds shot up to 47%.  When Bill Mueller knocked him home to tie it, the odds shot up to 73%.

There are two takeaways for the aspiring game designer.  The first is that the odds of victory (what Sabermetricians call ‘win expectancy’) can absolutely be quantified.  There are, in fact, iPhone apps that you can take with you so you can find your team’s win expectancy at the ballpark.  The second, more important takeaway is that that the low win expectancy of the Red Sox, that 23% number, is what made the game great – a game that, out of hundreds of championship baseball games that have ever been played, is still discussed today. It is, quantifiably, the underdog triumphing over long odds.

A graph of the win expectancy of Game 4 Red Sox-Yankees 2004 ALCS over time.  Note the spike when Bill Mueller singles in the run in the 9th inning.

Games and Win Expectancy

Movies play with win expectancy all the time.  When Darth Vader is trailing Luke in the trench near the end of Star Wars and breathes “I have you now”, you feel like Luke’s chances are hopeless (perhaps sabermetric geeks would say sub-10%).  But then Han comes in like a cowboy and clears Luke for the shot.  Han’s appearance is epic, but it is the emotional low beforehand that makes his appearance so fist-pumping.

Overcoming great odds is just one of the scenarios that can provide a unique emotional high to the player, and that can make a gaming experience even more memorable and compelling.  However, game designers have some real problems when attempting to manipulate win expectancy.  The first is the role that player skill plays into the equation: a 90% win expectancy for a seasoned Ninja Gaiden player may be only be 10% for the casual.

The more pernicious problem is that underdogs lose most of the time.  Game 4 is unusual because the Red Sox, statistically, should have lost.  But if a boss fight has an actual win expectancy of 10%, this suggests that the player should flat-out fail 9 attempts out of 10.  In practice, most players will throw their controllers through their flatscreens in frustration.  Still, those who have finished a boss fight with a sliver of health left know that the emotional peak that results is often worth the frustration that led us there.

On top of all of this, a win expectancy that is too high may indicate a game that is ultimately not very interesting.  A game where victory is effortless is one where the player fails to get emotionally invested and, as such, it’s very easy for him to put the controller down.

The Illusion of the Underdog

So we want players to feel like they are riding on the edge of failure, without actually putting them there.  This isn’t insurmountable  – the movies do it all the time.  Cognitively, you know that Luke is going to blow up the death star, but in the midst of the roller coaster ride, you’re fooled.

One easy way is to conjure the illusion of toughness – most commonly this is done with puffery.  Scaling up the boss in size, giving him an intro speech, giving him extremely showy particle attacks, or even doubling the size of his health bar can all instantly create the sense that you’re expected to fail.  Another is to simulate attrition, even fake attrition.  Star Wars does this by winnowing down Luke’s wingmen one by one.  In games, this can be done by knocking off a player’s armor, breaking his weapon, or killing a companion.

Most games, like the God of War series, set up boss fights with a good amount of trash that can easily be overcome.  Just as a low win expectancy can make the resulting victory that much higher, having an easy baseline experience can make a boss feel much tougher than he actually is.

Playing with the player’s health bar is another way to create an illusion that the player is closer to death than he actually is.  In Diablo, a boss creature can knock a player’s health way down – players can easily resolve this by quaffing a potion, but seeing that health bar go that empty so quickly dramatically adjusts his perceived win expectancy downward, while still giving him good statistical odds to recover if he plays correctly.  The health bar can also unlock functionality: tanking Paladins in WoW get access to a talent that results in them taking less damage when they are below 35% health (which helps put them in that ‘danger zone’ more often), and some fighting games give players access to super attacks that are only available if they are below a certain health threshold.

Player vs. Player Activity

Most single-player games are designed, nowadays, to let the player win – they might have an overall win expectancy for most of their content at about 95%.  However, this is not the case in player vs. player scenarios.  In a one on one StarCraft match, for example, a player’s chances of winning start at 50% – and that assumes the players are evenly skilled.  Add contestants, such as in a Quake Deathmatch or a large game of Risk, and the individual’s chances drop with each additional contestant.  This is especially problematic in games where both kinds of content exists – a PvE player in an MMO like World of Warcraft or Warhammer Online who is used to winning 95% of the time may have trouble adjusting to the sudden increase on failure on their part.

This is one of the reasons that team gameplay is important – having only two teams in Capture the Flag means a Quake player is much more likely to experience victory than in a Deathmatch.  Furthermore, to some degree, it disguises a player’s own contribution to failure.  If you lose a one-on-one match, you have no one else to blame.  If you lose a four-on-four match, that sense of failure is ameliorated.  And sometimes, a low-skill player will experience victory by osmosis.

Calculating win expectancy in situations with asymmetric power can be extremely complex.  In the board game Illuminati, the Gnomes of Zurich are by far the most powerful Illuminati faction, but according to Steve Jackson, they don’t win disproportionately often.  It turns out, most other players recognize that power, and gang up on the hapless sap playing the gnomes.

The Illusion of Closeness

The polling for the presidential election of 2004 was never even close – nearly every poll in the election gave George Bush leads over John Kerry .  This was even more true in 2008 – other than a short spike around the Republican convention, Barack Obama had convincing leads over his opponent John McCain from the moment that he secured the nomination.  In both cases, the frontrunner played defense, avoided making any mistakes, and secured the election with relatively little doubt.

You wouldn’t know that from the election coverage.  Every news story made it clear the underdog was within arm’s reach of the frontrunner, and that the election was anyone’s game.  Close elections sell more newspapers than blowouts.  The candidates were complicit, too.  The frontrunners wanted the election to feel close to create pressure on their base to turn out.  The underdogs have to seem to have a fighting chance if they want to attract donations and volunteers.

Close matches are compelling because the expectations around a contest remain in doubt.  In blowouts, there is precious little doubt, and the contest ceases to be interesting.  Ensuring that there is doubt, that there is always an angle for an underdog to get back into the thick of things, is a time-honored design principle.  Consider Mario Kart: there are a lot of powerups in the game that allow you to attack other racers on the track, but by and large most of them fire forward.  One well-aimed tortoise shell can knock a frontrunner to the back of the pack.

Having slightly opaque game mechanics can also create this doubt.  In Settlers of Catan, the player may purchase cards that offer secret victory points.  The leader may have a convincing lead over other players, but anyone who has cards in his hand may actually be able to surprise the leader and steal victory, which maintains interest in the game.

The Future of Win Expectancy

Game designers are already becoming more and more sophisticated about building a good win expectancy cadence in their experience – ensuring that players alternate between assured victories and more challenging content.  I expect we will see more sophistication of this nature in the future.

Already, games are starting to notice that players are failing more often than expected, and asking them if they want to adjust their game’s difficulty on the fly.  In the future, more will start doing this behind the scenes – to have AIs determine the worthiness of their foe, and adjust automatically.  This is already commonplace in racing games like Kart Rider, although designing systems to do so without cheapening the accomplishments of your more hardcore players takes more design thought than you might expect.  Left 4 Dead similarly uses an ‘AI Director’ in order to provide an appropriate level of challenge to whichever four players have found their way into a match.

Game mechanics like Xbox Achievements and Boasts in Fable are already helping players to make this choice manually.  More casual players can just play the game as presented, and experience a default win expectancy that was balanced for them.  Hardcore players who can blow by that challenge as-is can choose to make the fight harder: in Fable, players can boast they can defeat a boss unarmored, or without taking damage.  In World of Warcraft, players can earn achievements for killing bosses without losing any raid members, within a time limit, or in other ways that provides hardcore players a challenge without bending the difficulty curve to a place where more casual players can never conquer it.

Surprising players is important – they need to expect the unexpected.  But it is equally important for game designers to cater to what players do expect.  Doing so will give the players greater understanding of their own failure, and increase their satisfaction when they succeed.  Win expectancy is a powerful tool – designers should be willing to cheat and manipulate it in order to provide the appropriate sense of challenge and tension in their game experiences.