Why Games Must Be More Than Fair: Random Walks,
Long Leads, and Tools to Encourage Close Games
Kevin Gold
Wellesley College
106 Central St.
Wellesley, MA
[email protected]
ABSTRACT
It is natural for a game designer to assume that games that
give each player an equal chance of winning will often have
tie games and changes in the lead, but this is incorrect.
An argument is made for viewing fair games between players of equal skill as unbiased random walks, the result of
which is that fair games naturally tend to have few or no tie
games and lead reversals, unless they possess mechanisms
that specifically encourage close games. The mechanisms
of Negative Feedback, Different Speeds, Rise and Decline,
Increasing Stakes, Bounded Success, and Alliances are presented with real game examples as ways to avoid blowout
games, and the conditions of their success are evaluated in
light of the random walk model. Examples from well-known
games and potential recombinations of these mechanisms
with new genres are discussed.
Keywords
Game design, probabilistic models, random walk, fairness
1.
INTRODUCTION
primary point is to drive home the idea that to have close
games, game design cannot be merely fair; it must somehow
drive the players’ scores closer together.
The paper will begin with some mathematical results about
long leads that have been long known to mathematicians of
probability, but deserve to be better known to game designers. It will then point out several approaches to encouraging
more close games that have been tried in the past, categorized here as Negative Feedback, Different Speeds, Rise and
Decline, Increasing Stakes, Bounded Success, and Alliances,
and explain in terms of the random walk model why and
when they are effective in producing games with more reversals of the lead. Most of these techniques have only been
historically applied in a few genres, but could easily be applied to others. With luck, this paper will get game designers
thinking about new techniques in their preferred genres to
encourage close games.
2. THE RANDOM WALK MODEL OF A FAIR
GAME
One would expect that a fair game between two completely
evenly matched players will tend to go back and forth between them – and one would be wrong. Instead, even in a
two-person coin-flipping game, where the Heads player wins
if the most Heads are tossed, half of all games played will
have no ties during the second half of the game [12]. Similar
results shall be described below that show that players who
are ahead in fair games, tend to stay ahead the whole time –
running counter to our intuition that games between equally
matched players should go back and forth.
What follows is a brief introduction of the concept of a random walk.
Game designers would usually like their multiplayer competitive games to be exciting, with possibly several reversals or
ties before the end of the game. The designer may assume
that if a game provides equal opportunities and abilities to
each player, that such reversals will occur naturally between
players of equal skill. This is not the case, and this paper’s
We can plot the difference between the two players’ scores
over time (Figure 1). This value will start at zero, as both
players begin with equal scores. With probability 1/2, player
A will score the first point, and the next value will be +1;
with probability 1/2, player B will score the first point, making the difference -1. After that, either path has a probability 1/2 of returning to 0, and a probability of 1/2 of further
deviation. Two total paths return to the origin out of the
four that are possible at the second time step, so the probability of a tie when the second point is scored is 1/2.
Suppose two players of equal skill are playing a game in
which it is possible to score points, and their ability to gain
more points is independent of the current score. Because
there is no reason to prefer one player over the other for
believing who will get the next point, it makes sense to say
that each player might score the next point with probability
1/2.
The path from the origin that the difference in scores takes
over time is known as a random walk. Every time the walk
touches zero, this means that our two players have a tie
game. Every time it crosses from one side to the other, it
means that a player has come from behind to take the lead.
Score Difference
The following theorems and corollaries about random walks
are all from [12], and adapted somewhat to game terminology. Time is here indexed by whenever a player scores a
point – in other words, each step of the random walk. The
times in question are often given as multiples of 2 because
ties can only occur with an even number of points.
A winning
Theorem 1. (Arc sine law for last visits.) The probability that the last tie in a fair random walk game of length 2n
occurs at time 2k is given by:
α2k,2n = u2k u2n−2k
(1)
tie
u2k is the probability of a tie at time 2k, and is given by:
B winning
Points Scored
u2v =
Figure 1: Modeling the difference between scores of
two equally skilled players as a random walk.
If the walk never returns to the origin after a certain point,
it means that whoever was in the lead, remained in the lead,
until the end of the game.
We can use this walk to model games that do not have points
as well. The units on the y axis can be goods, or power of
overall board position. The game does not necessarily need
to be random itself; we can still treat whether one player is
winning or the other as random, if the players are of equal
skill. It also serves as a model of games that move through
physical space or virtual space – a first-person shooter such
as Halo [4], for example – as long as discrete events occur
that signal that one player has gained or lost an advantage.
Though this model assumes that the probability that either
player scores a point is the same, in practice, players may
have different skill levels, and the probability that the walk
goes in one direction will be at least slightly greater than the
other. However, the results proven here will be all about how
unlikely it is for this walk to return to the origin, and so the
assumption that the players are of exactly the same skill level
is a “best-case scenario.” We shall see that even in this bestcase scenario, ties are uncommon, and the lead is unlikely to
change much over the course of the game. Similarly, in many
games, doing well leads to doing even better in a positive
feedback loop; this is true, for instance, in any game where
resources can be used to purchase tokens that beget more
resources, as in a standard real-time strategy (RTS) game, or
in a building-themed Euro board game such as The Settlers
of Catan [28]. Again, the fact that our pessimistic results
hold even in the absence of such positive feedback loops
only goes to further prove the point that close games require
active intervention on the part of the designer.
3.
PESSIMISTIC THEOREMS ABOUT FAIR
GAMES
!
2v −2v
2
v
(2)
Corollary 1. With probability 1/2, no tie occurs in the
second half of the game, regardless of the length of the game.
The proof of the theorem can be reached by counting the
number of possible paths for the random walk to take that
touch the origin at time 2k, multiplying by the number of
paths that do not touch the origin after that, and then dividing by the total number of paths that run the length of
the game. The corollary follows from the fact that the equation is symmetric, with the value for 2k equal to the value
for 2n − 2k, so that the probability that 2k > 2n/2 = n is
equal to the probability that 2k < n.
The plot of Equation 1 for different values of 2k is always
U-shaped, suggesting that the most likely times for the last
tie in the game are either right at the beginning or right at
the end. The latter results in a nice climax for our game, but
the former results in a game during which the lead hardly
changes at all, and these two occurrences are equally likely.
Theorem 2. (Discrete arc sine law for sojourn times.)
The probability that a fair random walk game that ends at
time 2n spends 2k time units in one player’s favor and 2n −
2k time units in the other player’s favor equals α2k,2n .
Recall that the plot of α2k,2n is U-shaped, so this theorem
says that the event of two players spending the same amount
of time in the lead is the least likely outcome. The most
likely outcome is that one of the players dominates for nearly
the entire game. The proof is somewhat long and omitted
here.
One final result from [12], and its corollary. A change in the
lead occurs when the player who is winning at time n − 1 is
different from the player who is winning at time n + 1.
Theorem 3. In a fair random walk game, the probability
ζr,2n+1 that up to time 2n + 1 there occur exactly r changes
in the lead is given by:
ζr,2n+1 = 2p2n+1,2r+1
(3)
where px,y is the probability that the score difference is y at
time x.
Corollary 2. The probability of r reversals of the lead
decreases with r and is greatest at 0.
The proof of the theorem is another path-counting argument, which maps paths that cross the origin to paths that
reach certain score differences; see [12] for details. The corollary then follows because more extreme score differences are
less likely than close score differences.
Thus, even though we are modeling a game between two
players who are equally likely to score points, the most likely
number of changes in the lead is 0. This is even in the
absence of differences in player skill and positive feedback,
which are even more likely to contribute to games in which
the lead does not change. Therefore, to encourage games
with multiple ties and reversals, the game designer must take
a more active role than simply providing the same chance of
winning to all players.
4.
ENCOURAGING CLOSE GAMES
Now that we have covered the theorems that suggest pessimism about the prospect of fair games resulting in close
games, we can turn to various techniques that game designers have employed to encourage reversals of the lead. We
shall prefer qualitative over quantitative descriptions of how
these fit in with the random walk model, since the game
designer usually has control over too many variables for the
equations to be anything but messy.
4.1 Negative Feedback
A natural control approach to the problem of scores drifting
apart is to push them back together. If the probability that
the losing player catches up is made to increase as the score
difference increases, then the scores will tend to not drift
quite so far apart. Reducing the chance that the random
walk wanders too far from the origin increases the chance of
a tie, and therefore increases the chance of a score reversal.
Negative feedback is one of the more popular approaches to
the problem of runaway players. Corruption, for example,
was used as a mechanic in Civilization III [3] and the recent board game Through the Ages [8] to make very large
civilizations have a more difficult time producing goods.
Since civilization games often reward success with more success, as larger civilizations become more productive and
more able to grow, there comes a point at which growth
is slowed or reversed so that smaller civilizations have at
least some chance to catch up. Any game with expenditures
required for growth can increase those expenditures as players progress, though care must be taken to ensure these are
large enough to boost the player who is behind, instead of
simply prolonging the game. To avoid this dilemma, some
games in which players build on themselves simply allow
the unchecked growth and thereby end the game quickly, as
in Race for the Galaxy [23] or Dominion [29]; these games
typically have no reversals, but make up for it with brevity.
The other approach to negative feedback is to boost the
player who is behind, by making every successful attack from
a winning player increase the losing player’s resources. In
fighting games such as Street Fighter IV [7], this is done
by allowing a player to unleash a devastating attack once
the player has taken enough damage from the other player.
Super Puzzle Fighter II Turbo [6] and other competitive
puzzle games make a player’s attack consist of gems that,
after a certain amount of time, become usable by the attacked player for their own comeback. Super Puzzle Fighter
II Turbo took this particular idea furthest by having the
dropped gems obey patterns that could be learned for each
character, maximally increasing the chance that the attacked
player could use them for a reversal. James Ernest’s The
Great Brain Robbery [11] gave experience points to players
whose zombies lost their valuable brains in a fight, explaining, “Experience is what you get when you don’t get what
you want.”
The downside of using negative feedback is that it can be
complex to implement. In a board game, using multiple
costs for items adds extra calculation that the players must
perform manually. Video games lack this problem, but the
approach still requires more design tinkering than other approaches, since when to apply feedback and how much are
extra parameters that must be adjusted during playtesting.
The more complex the feedback formulae, the more playtesting required to set each constant.
4.2 Different Speeds
Mike Flores wrote an influential article about Magic: the
Gathering [15] called “Who’s the Beatdown?” in which he
argued that in every Magic matchup, one player is the Beatdown, who wants the game to end quickly, and one player
is the Control, who wants the game to go on long [13].
Traditionally, a Beatdown Magic deck has resources that
are cheap and fast, but which quickly become useless; the
Control deck contains cards that become better with age,
slowly building the player’s power, but presents no immediate threats. Flores’s point was that even when two Beatdown players face off, one player is secretly the Control and
doesn’t know it, because no two decks peak on the same
turn, and one deck will fare better in long games than the
other.
It is not common for Magic games to have multiple reversals
of board position, but it is common to have one, when the
Control player passes the Beatdown player. In the random
walk model, we can think of our the walk as being over the
net difference in power between the two players, with the
Beatdown advantage arbitrarily set to be the positive side.
The walk will be biased at first toward +∞, with a greater
chance of success for the Beatdown player, but gradually
this bias is reduced and then reversed toward −∞, with the
Control player having a better chance of improving board
position advantage. Because the walk was very likely to
start off heading toward +∞, but the asymptotic behavior
of a negatively biased walk is toward −∞, the walk is very
likely to cross the origin once.
Of course, it is possible for the walk to never cross the origin at all if the Beatdown player wins. But the players often
have knowledge about what would constitute a reversal of
the bias in the game; for instance, certain combinations of
cards for Control produce “locks,” in which the other player
can do next to nothing, and getting close to establishing
such a lock is good enough to produce tension in the game,
even if the Control player has very little power until the
lock is established. For games between a Beatdown player
and a Control player, the correct measure of tension is not
necessarily the board position itself, but approaching the
point at which the bias shifts toward the Control player –
approaching the zero-crossing of the derivative of the players’ power difference, rather than the power difference itself.
The game remains interesting even if the Beatdown player
is far ahead, because both players know that the advantage
is eroding; and if the Beatdown wins without even nearing
the zero-crossing of the bias, at least the game was short.
Magic achieves this dynamic because no two decks peak on
exactly the same turn, and because varying resource costs
to play cards allow designers to make some cards that peak
quickly, while others peak slowly. This kind of asymmetric
design is applicable to several genres – for instance, the famous “Zerg rush” of Starcraft [24] was a tactic by a Beatdown
race to destroy the Control races before they had a chance
to stabilize. Whenever asymmetric design gives one player
cheap resources and another expensive resources, there is a
chance to encourage a reversal. Others have interpreted the
practice of asymmetric design in real-time strategy (RTS)
games as adding interest through a game theoretic choice
[26], but perhaps the true contribution of the asymmetric
races pioneered in Dune II [2] and perfected in Starcraft
was that they were a solution to the problem of reversals in
a genre in which success breeds success.
The downside to the Different Speeds approach to encouraging reversals is that there is nothing to sway the game back
to the Beatdown player once the game is in the hands of the
Control player. This can lead to some truly tedious games;
Richard Garfield relates an anecdote in which an early Control player won by playing cards that removed all threats
and defenses from the opponent’s deck one at a time, until
the opponent could do nothing but take a single point of
damage each turn until death [14]. Enforcing one reversal is
better than no reversal at all, but it is perhaps not the best
solution for an interesting game.
conquers a large amount of territory.
In terms of the random walk model, this is the natural fix
for the Different Speeds model: the bias shifts back and
forth. As a civilization is played out, the random walk is biased more and more against a player, until finally the player
must send it into decline, further increasing the negative
bias. But with a fresh civilization, the bias swings strongly
in that player’s favor again. Whether this shift is enough
to drive the game into a full reversal, or simply creates a
more meandering path away from the origin, depends on
the specifics of the game.
Cooldown timers in massively multiplayer role-playing games
(MMORPGs) such as World of Warcraft [1] arguably serve
a similar function. While an ability is inaccessible while the
player waits for the cooldown timer to expire, the character is essentially in decline. Abilities with cooldown timers
are more likely to engender score reversals if the timers are
asymmetric between players, because players that peak at
the same time are essentially not changing the game balance at all, but merely increasing the stakes (see Increasing
Stakes, below).
Taking turns going first in a game where resources can be
claimed is also a way to implement Rise and Decline; Puerto
Rico [27] does this, for example. The first player to go first
obviously has an advantage, however, unless all players get
an equal number of turns. Care must be taken to ensure
that all players get a roughly equal number of cycles, or the
game will be biased toward players who peak earlier.
A related danger with Rise and Decline is that the oscillations must not be so great as to make the end of the game feel
arbitrary. Risk [22], for example, implements both Rise and
Decline and Increasing Stakes (see below) with the periodic
trading in of cards for more and more armies, but the oscillations can be so great as to make the final victor seem to be
somewhat arbitrarily determined by who happened to draw
sets at the right time. Risk 2210 [9] and Small World both
address this problem with a fixed number of game turns, so
that the players can best plan their final rise; but this comes
at the cost of forcing the players’ final rises to sync together,
making a climactic change of the lead less likely.
4.4 Increasing Stakes
4.3 Rise and Decline
Since returns to the origin are more likely from points closer
to the origin, another approach to make reversals more likely
is to effectively reduce the number of moves it would take
to return to the origin as the game progresses. Increasing
stakes can make the origin closer by reducing the number of
moves it would take to get there.
The natural extension and fix to the Different Speeds approach is the Rise and Decline approach, in which the power
of players waxes and wanes cyclically. “Declining” a civilization is an action one can take in the board game Small
World [21], a reimplementation of the board game Vinci
[20]. When it appears a civilization has run its course, a
player can choose to send their civilization into decline by
skipping a turn; thereafter, the civilization is highly vulnerable. In exchange, the player can enter the game the following turn with a fresh, powerful civilization, which quickly
An extreme example is Jeopardy! [17], in which players can
bet up to all of their earnings in the the Final Jeopardy
round. As long as the winning player’s earnings are not more
than double a losing player’s earnings, the losing player with
equal skill has a good chance at a reversal – at least p(1 − p),
if the probability of knowing the answer to a question is p.
This is much higher than the chance of reversal from another
round, starting from unequal scores. Poker also benefits
from this possibility of a player going “all in.”
The curious thing about the random walk result, however,
is that even with increasing stakes, the difference in scores
is likely to wander away from zero and never come back on
repeated trials. Increasing the stakes increases the size of
the jumps in score, but changing the units on a random walk
does not change the results mentioned earlier. If the game
continues after a rise in stakes, it is very likely to continue
with winning players winning more, and losing players losing
more, making this an ineffective strategy for reversals unless
used for a final “all in” climax.
4.5 Bounded Success
If the trouble with a random walk is that it is likely to
wander away from the origin, then perhaps it makes sense
to bound the walk so that it cannot get too far from the
origin. An extreme example of this is to clamp the walk
at one jump away from the origin, so that it the game is
always at most one point away from a tie. This seems to be
a good recipe for exciting games, if the score is always close
by design, though it seems as if the winning player may not
have much to do.
Curiously, it is some of the oldest children’s games that exhibit this behavior, and not usually modern video games.
Tag is a game of bounded success and bounded loss; a player
is, at best, not “it,” but this position requires constant vigilance, and a player is at worst “it,” but only a touch away
from “winning” again. King of the Hill games in which the
goal is simply to defend a position from anyone else are similarly bounded; one is either the King of the Hill, or not.
Curiously, Halo [4] did away with this nice property of King
of the Hill in its implementation, by defining success as holding the Hill for a specified amount of time before declaring a
win. This made it very likely by the results mentioned earlier that one team would stay ahead in total time for most
or all of the game. Games of Bounded Success run contrary to the competitive gamer’s intuition that games with
score are better because they measure the skill differences
between players. First-person shooters (FPSes), especially,
tend to incorporate no mechanisms at all for encouraging
reversal, and this decision may be driven by gamer perceptions of these games as highly skill-driven. One can hypothesize that it was this perception that caused the Halo
designers to add a scoring system to a game format where
there has traditionally been none. The long leads that result
from random walks can thereby introduce the illusion of a
skill-driven game even in a very random format. This is an
example of the “hot hand” illusion, in which people falsely
attribute random winning streaks to temporary changes in
prowess [16].
Regardless, not all game players play to measure their skill,
and players who play for some other reason – and there can
be many [18] – might be well-served by games of Bounded
Success.
4.6 Alliances and Kingmaking
Finally, there is always the prospect in multiplayer games of
players ganging up on the person in the lead. Munchkin [19]
and Bitin’ Off Hedz [10] are two examples of games in which
players are given resources that have the specific purpose of
bringing down another target player, typically the player in
the lead. Games that do not contain any overtly aggressive
elements can still see alliances form against players in the
lead: trade embargoes on players in the lead are familiar to
players of The Settlers of Catan [28], for example.
Thinking about the pairwise random walks of player score
differences, we notice that ad hoc coalitions made to damage
the lead only affect differences with the player in the lead;
random walks of score differences between any two players
who are not close to first place remain unaffected – which is
to say, they are somewhat less likely to pass each other than
we would like. The top two players’ scores are likely to be
drawn closer together, as players target one, then the other,
then the first again, but unless the players’ targeted actions
are truly brutal in their effects, no other two players’ rankings will be affected. This has the unfortunate corollary that
coalitions do nothing to change the rankings of the players
who are not close to first, which are likely to remain the same
the whole game. This leads to the phenomenon known as
“kingmaking”: last place players who never got out of last
place can still expend resources to damage players near the
lead, and the winner can be decided by these disgruntled
players.
On the whole, ad hoc coalitions to damage winners do not
do much to make races more exciting, because to the extent
that they increase reversals of first and second place, it is
largely because of the decisions of neither of those players.
Games in which multiple players can win, on the other hand,
such as joint-win Diplomacy [5] and Cosmic Encounter [25],
are possibly a different story, since the existence of such
alliances makes multiple players just one deal away from
victory.
5. CONCLUSIONS
An analysis of random walks suggests that the problem of
blowout games, in which the lead does not change, is common even in chance-based games or games between players
of equal skill. A game designer who simply believes that
player symmetry or a high degree of randomness will lead
to close games is in for a surprise on playtesting. Luckily,
game designers have come up with many ways over the years
to make games closer and more exciting. To a large extent,
this has often been done with an eye toward allowing players of different skill levels to play against each other, but the
techniques can also be used to ensure that any competitive
game is as close and exciting as possible.
Creatively applying the designer’s toolbox for dealing with
problem of long leads can lead to approaches that have not
been tried in particular genres. Why not have a fighting
game with Rise and Decline, where players must choose
when to go into decline and be vulnerable in exchange for
power later? What about a competitive puzzler with Different Speeds, where players can select between fighters that
require fewer gems for matches (Beatdown), and those whose
attacks scale more dramatically with larger matches (Control)? Can we design a casual RTS with Bounded Success?
The tension and excitement of the competitive game is highest when the game is closest, and an analysis of random
walks suggests that the game designer should take an active hand in encouraging close games. There are many approaches that have not been tried for any given genre, and
perhaps contemplation of the random walk can suggest still
more approaches.
6.
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