In quantum or classical mechanics, there’s a well-known class of problems that has traditionally proven very difficult to solve, complex problems that model the motion of three particles or objects such as the sun, earth, and moon: the “three-body problem.”
In games, the equivalent of this is the “three player problem.” There are actually two problems. In a three-player conflict game without a definite end, the two players who are behind will usually beat on the one who is ahead, resulting in a perpetual stalemate. This can be avoided by placing a time limit on the game. However, as soon as you add a definite end to the game, such as a number of turns or a number of points, the situation changes drastically. Perpetual stalemate is no longer possible. But it is frequently possible for one player (call him “A”), if he believes he will lose and cannot catch up in the remaining duration of the game, to determine which other player wins. That is, late in the game the losing player exerts all his efforts against another player “B,” which tends to let the third player “C” win. R. Wayne Schmittberger (long-time editor of Games magazine) called this the “petty diplomacy problem” in his book New Rules for Classic Games. “Petty” here is derived from “pettiness,” I believe, implying that the player who thinks he’ll lose often chooses for petty reasons which of the others to take with him to defeat.
This is not usually a problem in games where players can do little to affect other players, such as most race games. If a player can hardly affect the other players (as though each were playing alone), then the “petty diplomacy problem” goes away. Many “Euro”-style board and card games (which are often for three or four players) have been called “multiplayer solitaire,” a popular style partly because it avoids the “petty diplomacy” problem.
But it can be a big problem in wargames and other conflict games. And this is why we rarely see such games that are “naturally” good for three players. Generally most conflict games are best for two, or four to five. When there are four or more players, a single player has less influence on the outcome, mitigating the effects of “I’m going to lose so I’ll make sure you don’t win.”
I’ve discovered a number of ways to retain a strong competitive aspect in a game yet make it work well for three players. The example games I’ll describe are non-electronic but these solutions apply just as well to videogames. I’m going to follow the examples with a discussion of problems that occur in any game with more than two separate interests, all of which can be applied to the three player problem.
The general description of the two solutions is:
The victory condition must be one that can be attained in a single move/turn, and that can change over time, yet players can usually (but not always) anticipate that it might occur. This enables two players to “gang up” on the third to stop him. If they succeed, the game returns to an equilibrium until one player once again threatens to win. Even without a time limit, the game will eventually end.
Have a situation where changes in the state of affairs are quite small, so that no single move can make a very large difference in the state of things, in who is ahead and who behind and what the margin is. Only late in the game will a player be able to recognize that he’s not going to win. By that time, he usually cannot make enough difference in the short run that remains, to throw the game to another.
Actual examples will make this clearer.
Law & Chaos
In my prototype titled Law & Chaos (in the publishing queue with Mayfair Games, and likely to have the name changed), I serendipitously discovered how the first method can work.
The game is played on a simple hexagonal board (61 hexagons) with one kind of piece. The game is an example of one that arises from components, as I wanted to devise a way to use in a game the colored glass beads or “stones” so common in flower-arrangements nowadays. (In the end, I had to abandon those pieces because they weigh too much, as publishers usually pay for shipping to distributors!)
Unadorned hexes and one kind of piece don’t offer much room for variety, and I settled on using cards to provide that variety. There are three types of cards:
Victory conditions. These specify patterns a player can form to win the game, such as a row of six stones
Capture methods. These specify patterns that let a player capture opposing stones
Action cards. These enable the player to take/cause a specified action (such as preventing the replacement of a victory condition card)
Each player has cards in hand, and there are cards on the table that apply to players selectively. Each turn a player either places a piece on the board (which could also result in one or more captures) or plays a card. The key for our discussion is that anyone can win at any time by having the right victory card in combination with the corresponding pattern on the board. The card needs to be face up on the table already, or it must be played from the player’s hand; in the latter case the other players have one round to try to break up the pattern. (There are more details than this, especially about player interaction, that will await the game’s publication.)
Players soon learn the possible victory patterns (which are also illustrated on a separate sheet), and can either see that another player has the right victory card on the table for a pattern he can soon form, or can anticipate that it is in the player’s hand, as that player builds patterns.
When the other players see this, they tend to gang up on the “leader” to thwart his plan. If they succeed, then they play once again in a kind of equilibrium. If one player is far behind (it’s possible to have no pieces on the board), the others will usually concentrate on one another, which gives the third a chance to catch up. But sometimes the other players will fail to see that the “leader” is threatening, or will be unable to stop him or her.
In the end, it is that rarity, a game that is not only good but outstanding for three players, and best played with three, though it can be played by two or four.
The “sudden victory that can often be forecasted and prevented” characteristic is the key. Unlike many wargames, some information must be uncertain or hidden, or players will always know who is threatening to win. It is also important that the victory conditions themselves can change over time.
Any kind of foreseeable/definite time limit helps “petty diplomacy” rear its ugly head. Using a criterion of victory by reaching a set point total, very common in boardgames these days, is unsuitable because the end is fairly predictable. On the other hand, a margin victory criterion– for example “have 20 points more than the next highest player”--means no one knows for sure when the game will end, because the players behind can catch up and the margin can get smaller as well as larger. When the player who is furthest behind can still hope to catch up, he’s not likely to give up and “suicide” against another player. If a set total of points determines when the game ends, then at some point the player furthest behind will realize he doesn’t have time to catch up, allowing “petty diplomacy” to kick in.
So this kind of game needs an open end, probably with some mechanism that prevents the game from going on forever but does not arbitrarily end it. In Law & Chaos I added that mechanism when playtest games among expert players stretched to more than two hours, much longer than the average. The mechanism comes into play at a certain point, by increasing the number of ways players can win.
The game doesn’t work as well with two or four players. With two, when a player “gets it going,” there’s a steamroll effect that’s hard to stop. With four, it’s too easy for three to gang up to stop one, and that led to a variation of the game, recommended for four players, that limits what a player can do to stop someone else because they do not have a hand of cards.
A military version of this kind of game
How might this apply to a military game rather than an abstract one? Occasionally we see three-way, rather than two-way, historical struggles for supremacy. The one that first comes to mind for me is Athens, Sparta, and Persia in the fifth century BC. Athens and Sparta joined together to defeat the great Persian invasion of 480 BC. Then Athens’ sea empire became dominant for a while, and Sparta abandoned the alliance against Persia. By the end of the Peloponnesian War, Persia controlled Greek Ionia, and gave subsidies to Sparta, enabling the Spartans to build fleets to defeat Athens. In the fourth century it was Sparta’s turn to be pulled back, this time by Thebes in conjunction with Athens. Finally, the Macedonians conquered Greece, and soon after conquered Persia with the help of the traditional Greeks.
In other words, one power or another threatened to become completely dominant, but was defeated by the other two, until yet another more-or-less Greek power defeated them all.
How can this be represented in a game? The key will be a card for each round of play that will indicate a victory condition for one of the nations (or perhaps all three). If the condition is attained, the game will end right there. The order of play will change from round to round to help give nations a chance to thwart the one closest to a win, and some mechanism that abandons the move-all-units-in-your-turn tradition might be desirable as well. There’s also a possibility for a point game here, with the card specifying the points that can be earned by achieving certain objectives, and if any nation reaches a given point margin over the next highest, it wins. In essence, in many if not all rounds we want at least one nation to have a chance to win while the others must thwart that nation, possibly ignoring their long-term interests in favor of defeating the temporary leader.
Because there have been at least two Peloponnesian War games published in the past few years, I have not tried to develop this game.
Other Historical Wargames
Now what about the second version of a natural three player conflict game? I call this the “equilibrium” method. I have a prototype Britannia-like game called “Frankia: the birth of France and Germany,” that includes three scenarios, 406-814 AD, 843-1215, and 1215-1492. As in my game Britannia, each player controls several nations in the course of the game, and victory is determined by points. The first two scenarios are for four players, as with almost all Britannia-like games, but the last one is for three. I decided to try this because the situation in this scenario is so different from the typical Britannia-like game. The typical situation sees major raids and invasions from external sources, and a considerable turnover of nations and locations–which is the situation in the first two scenarios. In 1215-1492 in this part of the world, the nascent nation-states of modern Europe could already by discerned, there were no external invasions or raids at all, and in part thanks to geography, borders did not shift a great deal over time.
I was fortunate insofar as I had changed two of the fundamental systems of Britannia for Frankia, the combat system and the economic system. The three player scenario would not work as well with the combat system used in Britannia, in which casualties can be quite high. (Roll a die for each army in a battle; a 5 or 6 kills an opposing army.) Frankia uses a card-based, diceless combat system that does not eliminate more than one army in a fight, and often none at all. Sometimes, in fact, the battle ends with both participants jointly occupying the area they’ve contested.
The economic system depends on the number of armies a nation already possesses, so that Powers tend to reach an equilibrium until they can conquer more territory. Your economy must pay to maintain existing units before you can get new units. In contrast, in Britannia, the number of new units you receive depends only on the economic value of the areas you hold, regardless of how many units you already have. Armies are also a representation of population, so you can build up armies until you reach your population maximum. In Frankia, the cavalry and infantry armies are more specialized, not representing the population as a whole.
Finally, the number of units is relatively low, in part because the stacking limit, the number that can be in a given area, is lower than in Britannia. In particular, there is no unlimited-size stack. When you cannot concentrate large forces, it is harder to make a sweeping invasion.
The result is that this scenario is even less a “conquest game” than normal Britannia. Much of the time in Britannia, you protect what you have rather than attack, until your next big invasion comes into play. In Frankia, you don’t have the forces to launch a big invasion, nor can you kill the enemy fast enough to overwhelm them. It is a case of nibbling a little here, a little there, trying to be on top of an equilibrium. And that is what we need to help us avoid the “petty diplomacy problem.”
Another situation that lends itself to this kind of game is eighteenth century Europe. This was an era of “the laws of war,” a reaction against the excesses of the Thirty Years War. Professional, not mercenary, armies relied on long supply lines guarded by highly efficient earthwork fortifications. The “civilized rules of warfare” prohibited living off the land (the commonplace in the Thirty Years). The aims of warfare were usually limited to wearing out the opponent while gaining bits of land (and fortresses) here and there. It is another case of “nibbling”, though in the game ahistorical events can occur, just as in other games with this long chronological scope.
My prototype (tentative title, Struggle for Hegemony in Europe, 1689-1789) also sees each player controlling several of twenty-two nations–there are no uncontrolled neutrals. The game is for three to five players. The dice-based combat system once again rarely results in casualties, with much of warfare revolving around sieges and the occasional assault on forts. The lines of forts tend to prevent sweeping invasions. A submission rule allows small nations to become subordinate to an attacker (at a cost) rather than be wiped out. The economic system once again considers the number of armies and fleets you already have, and there are maximum numbers of units as well reflecting the general difficulty of maintaining large armies and fleets.
As “sweep of history” games, both of these have a definite number of turns, but that is tolerable for an “equilibrium” game. More changes occur in nation positions in this game than in the real world–the purpose is to have a good game–but it is largely a situation in equilibrium, which we need to avoid “petty diplomacy”. The scoring system also contributes to this. Prussia, for example, doesn’t score any points for areas in France, or deep in Austria, or in Russia or the Ottoman Empire. The Prussians may (rarely) invade France in order to reduce the French score, but gain no points from it themselves. If they’re pursuing their self-interest as indicated by their scoring areas, they are unlikely to stray far from home. The English might, in some games, conquer Spain with Portuguese help, but they won’t gain points for every area in Spain, any more than the French would.
The right situation, then, combined with appropriate combat, economic, and scoring systems, allows these two wargames to be played by three players while avoiding the petty diplomacy problem. It is still possible for two players to decide to try to wipe out the third, but when they choose to do that early in the game, the victim has the opportunity to throw his forces against one of his two tormentors with sufficient effect to throw the game to the third. The threat of doing so actually tends to help restore the situation to something close to equilibrium. Late in the game, with all that has passed before, the simple two-against-one is much less likely to occur, and will be less effective.
Of course, another possible way to avoid the three player problem is a wargame in which almost all information, including information about who is winning, is hidden from the players. If you don’t know you’re losing, or who’s winning, the petty diplomacy problem goes away. Then we have a kind of race, or something that wouldn’t be much of a game, in most cases.
Another Abstract
I’m going to go back to abstract games for a moment to describe another solution. What makes this game work for three players is that it isn’t really a conflict game, it’s a positional game. That is, while player interaction is high, you cannot throw or exert force against another player, though you may be able to temporarily eliminate one of his pieces. You can block a player from scoring, but this may help the third player more than it helps you.
The game is played on a four by four square grid with stackable pieces (tentative title, which is likely to be changed when a theme is attached, is Four by Four). To score you must get four in a row or four in a square–a little like Tic-Tac-Toe/Noughts and Crosses--and the scoring pieces are then removed. There are stack size limits depending on the number of players.
Some of the pieces have special powers, such as the “Top” piece that another player cannot place a piece on top of, and the “Order” piece which freezes the order of pieces in a stack. The pieces alone might not be very interesting, but each player also has a hand of cards. The sequence of play is that all play one card simultaneously, then play two rounds of placing pieces on the board. The cards have “initiative numbers” that determine order of execution both for the current card round and for the following placement rounds, so there is no question of consistent advantage for playing first the way there is with Chess and many other games. The cards allow various manipulations of the pieces, such as moving the bottom piece in a stack to the top or allowing you to move a piece (not necessarily one of yours).
Each time a player achieves the scoring condition he gets a point, and play goes to three points, or fewer if players agree. Sometimes when one set of scoring pieces is removed, another score is revealed, and it is also removed. It is not unusual for one player to be behind, but come back to win the game. As the game progresses, a player must often choose whether to prevent a score by another, or to work toward a score for himself. Late in the game you know who’s threatening to win, because you can see the score. The time of ending of the game is uncertain, but there is a mechanism to end it rather than go on indefinitely (when a player has no more pieces to play). The game can also be played by two or four, with rather different tactics in each case.
In some ways the three-player problem can be seen as a subset of problems that occur in many games involving more than two sides. I’ll discuss some of those next.
Problems with Multi-sided Games
Boardgamers call any game with more than two players “multiplayer”, but this term is used in the videogame world for “more than one player;” and in videogames, a “multiplayer” game often has just two teams, as in Team Fortress. So I use the term “multi-sided,”meaning more than two separate interests regardless of the number of human players.
Certain kinds of problems crop up in three player games just as they do in multi-sided games for more than three. These potential problems exist to a lesser or greater degree in most multi-sided games, but in the extreme become obstacles to enjoyment. These are:
Turtling
Leader bashing
Sandbagging
Kingmaking
Turtling
A player sits on the sidelines, avoiding conflict, while other players fight a debilitating war; then the turtle steps in and wins the game because the others are too weak to stop him. In videogames this is often called “camping”, although “campers” often choose the tactic because they can occupy a very good defensive position and kill many opponents without dying often. In boardgames, the player may avoid combat altogether.
How is turtling avoided? The clearest method is with a zero-sum game, such as Diplomacy. You can only gain units by taking supply centers from another player: the general definition would be, a player can only gain something that another player loses.
Most games are not zero-sum. However, if a player stands to gain more by attacking than by turtling, the turtle tends to fall behind. This requires that there be a positive rather than negative economy, that is, that a player can acquire additional force/capability over time through the game economy. This is what makes turtling somewhat dangerous in Risk, because the players controlling more areas gain more new units. Nonetheless, the entire system of gaining new armies through cards exists, in part, to encourage players to attack rather than turtle (if you don’t successfully attack in a turn, you don’t get another card).
In many tactical battle games, there is no economy (or, to put it another way, there’s a negative economy, as both sides gradually lose units). If there are only two sides, this is not a problem. If there are more than two, it becomes a big problem. Three player Chess, for example, is likely to be an exercise in turtling. Even if a player is awarded control of all remaining opposing pieces when he checkmates a king, this may not be enough incentive to attack rather than turtle.
Turtling can happen in a two-sided game. For example, in Chess a player may try to create a very strong defensive position and wait for his aggressive (or computer) opponent to make a mistake, then attack.
It is very common for beginning designers of multi-sided conflict games to allow, even encourage, turtling, because there is not a positive economy.
Another way to discourage turtling may be extreme uncertainty about the overall situation. The turtle needs to know when other sides have been debilitated to the point that he can attack and probably win. If he doesn’t have enough information to know when that occurs, he may decide he needs to go forward rather than turtle. But this is not a desirable solution, especially in a boardgame. It is more practical (and more often used) in card games, where the cards provide a simple, natural means of hiding information.
Finally, victory conditions can discourage turtling. In a game where points are scored periodically throughout the game, a player who turtles may not be able to score well. This will certainly be true if those point awards involve holding certain locations, or destroying numbers of opposing units. Even if the victory conditions only apply at the end of the game, as in Axis & Allies (where the objective is to hold enemy capitals, with no predetermined time/turn limit), the turtle is less likely to hold these areas at game end. Yes, if everyone else wears themselves out without achieving the victory criterion, the turtle may be able to sweep the board and then achieve the victory. In A&A, because there’s a positive economy as well, and because there are only two sides, turtling doesn’t happen.
Leader Bashing
Leader Bashing is simply the tendency to attack whoever is ahead. This is a necessary component of multi-sided conflict games, though generally absent from race games. It becomes a problem when the typical thing to do is to attack whoever is in the lead, regardless of one’s own position.
In my game Britannia and other Britannia-like games, there are two elements to discourage leader-bashing. First, players score at different junctures of the game, so it’s difficult to actually know who is ahead at a given time. For example, the Romans score a lot early on, because they conquer much of Britain. The (yellow) Romans can have more than a hundred points in turn five, more than any other color, but yellow would be well behind overall because the average Roman score is about 125. (The average score for each of the four colors at game end is around 217.) Experienced players understand whether the yellow color is doing better or worse than average, but even then, this must be seen in relation to the scores of the other three colors. As the game goes on through sixteen turns and seventeen nations, discerning who is ahead becomes more complicated. In other words, there can be honest differences of opinion as to who is actually ahead, and you’ll often hear players over a Britannia board each explaining (sometimes disingenuously) why they are behind and someone else is ahead.
Further, the latest version of Britannia includes scoring markers. If players agree not to track the scores on a scoresheet, the scoring markers provide further uncertainty about who is ahead. In a game like the Hasbro version of History of the World this kind of uncertainty is absolutely necessary to avoid rampant leader-bashing.
Second, in a four player game, if you expend your efforts trying to “stop the leader” at the cost of your own score, then the other two players benefit. In Britannia each nation has historical scoring objectives that sometimes conflict with other nations, but not necessarily the one that you want to “bash” right now. This doesn’t work as well in a three player game, because only one other player benefits, not two, but it certainly helps in Frankia and Struggle (described above) which use a simpler version of the Britannia scoring system.
Sandbagging
Sandbagging is the reverse side of leader-bashing. Sandbagging is pretending to be worse off than you are, to somehow disguise how well you’re doing. In some games in which leader-bashing is easy, that is, each player is able to exert some influence and exert it against any other player regardless of positioning on the board, then it makes sense to try to be slightly behind the leader near the end of the game. In a game such as Vinci this is a common strategy. Either you need to be fairly far ahead when the end of game approaches, or you need to be in second or third place in order to win when others bash the leader(s). (How this works depends on the players, of course: some refuse to attack the leader, concentrating only on how best to maximize their own score.)
The easier it is for other players to see the reality of the game situation, the harder it is to sandbag.
Kingmaking
Kingmaking is the more general term for “petty diplomacy problem”. If it is too easy for one player to affect another sufficiently to cause him to lose, regardless of the number of players, the game suffers for it. Obviously, as there are more separate interests, each player can have less effect on the game as a whole, and kingmaking becomes less problematic.
Of course, in games allowing negotiation, a player with a weaker position can try to persuade a stronger player to leave him alone because “if you attack me I’ll throw my entire force against you and you won’t win.” If players can significantly hinder one another, this kind of negotiation strategy cannot be avoided.
Further, in some games a player can influence any other, while in other games there are circumstances of geography or even of turn order that reduce the effect some players can have on certain others. For example, in Diplomacy (World War I), the English player can try to influence others to attack Turkey, but he cannot affect Turkey with his units until late in the game, because the nations are on opposite corners of the board.
As with the specific case of the petty diplomacy problem, if players don’t know that they’re losing, they’re less likely to try kingmaking.
Insofar as uncertainty tends to mitigate the petty diplomacy/kingmaking problem, the problem is less likely to occur in card games than in board games. Cards naturally hide information, whereas boards naturally reveal information. In this respect, videogames are often more like card games, naturally hiding information from the players.
In summary, here are some ways to deal with the three-player problem:
A race or “multi-sided solitaire”; players cannot do enough to hinder/harm another to make a significant difference in the end portion of the game
Sudden victory that can frequently be forecasted and prevented, with changing victory conditions
“Equilibrium”, no side can drastically alter the situation in a single turn
A game that is almost entirely positional
Extreme uncertainty about who is winning and losing
Britannia references:
Wikipedia: http://en.wikipedia.org/wiki/Britannia_%28board_game%29
Boardgamegeek: http://boardgamegeek.com/boardgame/240/britannia
Publisher’s Site: http://new.fantasyflightgames.com/edge_minisite.asp?eidm=42&enmi=Britannia