Thieves' Blotto
I've made another microgame. I may have a problem.
I've yet to test my last one, which might make writing about this new one seem premature, but it's surprisingly hard to recruit three or four willing playtesters to try a 2v2 hardcore information based card game – I was fortunate in previous locations to have a ready supply of cerebral board gaming friends, but I've yet to make contact with such people in Nottingham, meaning anything I do want to test in the near future should probably be less restrictive on the number of participants and their gaming preferences. I should definitely try to shift things more towards the cool experience end of the gaming spectrum (e.g. horror game), rather than the abstract mechanics end (e.g. signalling, deception, trading) – not only because I haven't really sunk my teeth into this style of design yet, but because it's much easier to sell most people on experiences rather than mechanics. Mechanics are what the games nerds and mathematicians among us love, and they're the unappreciated heart of gaming, but just like games nerds and mathematicians they don't usually make a great first impression. You've got to dress them up to get people interested.
However, this is a plan for the future. I mostly figured out this new game before I decided on the above, and it seemed a shame to waste it. Think of it as a last abstract hurrah before I plunge into the murky waters of flavour-first design.
This two-player simultaneous auction game doesn't yet have a proper name, but I've been referring to it as Thieves' Blotto, for reasons that should eventually become apparent. It's much more of a mechanical shell or component for a game than a complete game in it's own right, but hopefully it's still fun to play and at worst may provide some inspiration.
But first, maths!
The Art of Game Theory
In Jesse Schell's The Art Of Game Design (which I'd generally recommend), he presents the following opinion:
“You would think with a name like “game theory,” it would be of great use to game designers, but in truth, it can only handle such simple systems that it is seldom useful for designing real games.”
The author then goes on to discuss dominant strategies and why they're bad for your games, which I found deliciously ironic. Yes, game theory isn't directly applicable to a lot of game design, as it concerns the solving of small-scale idealised problems rather than creating large-scale interactive experiences, but the mechanical underpinnings of these experiences are rooted firmly in this branch of mathematics. Every time you ask your players to take an action to further a goal, you're asking them to solve a mathematical problem of varying complexity. People find solving these problems enjoyable, as long as the problem isn't too easy (obvious strategies) or too hard (irreducible complexity). Knowing how to solve these problems and quantify their difficulty is a more informed way of incorporating modular mechanics into your games, which should cut down on the amount of time spent implementing/testing poor iterations. I'll accept that most designers lean more towards inspired artist than analytical scientist, which may explain Schell's statement, but the dismissive tone irked me.
As a result, I thought I'd spend a paragraph or many going over some introductory game theory, in case it's of use to any nonmathematical designery types reading this (hello! maths is fun! why yes, I have locked the door! stop struggling and sit down!).
Nash! (.....aa-aah!)
Any one-stage simultaneous move game can be expressed as a payoff matrix – this is just a table with player actions along the axes, and rewards for each player in the body of the table. It's probably easier if I show you:
[[ The first number is what the row player wins, the second number is what the column player wins. The amounts don't have to be money, but I figured it'd be easier to understand that they're rewards if I made them such ]]
Most board games don't fall into this category, because one-stage simultaneous move games aren't particularly fun – they require a lot of analysis, there's one action, and then the game is over. They are, however, useful components of a lot of games – for example, in Cuba, the bidding mechanic used to vote in parliament requires all players to choose a number in secret, then reveal them simultaneously to determine some outcome. In fact, even multi-stage sequential move games (most board games) can be expressed as a (very large) payoff matrix, and the same techniques applied to them (though this is usually very hard and unwise in practise).
A pure strategy for a given player would be that player picking a certain one of their available actions – for example, in the above table, the row player could choose the pure strategy of them always playing action A. A given pure strategy dominates another pure strategy if the player can guarantee a better payoff from playing the first strategy over the second regardless of what the other players do (e.g. above, “A” dominates “B”). If cake is better than death in every situation, cake dominates death. Schell correctly points out that you don't want dominated strategies in your game, as they're never useful and only serve to waste the player's time considering them. However, it's important to note that not all pure strategies dominate or are dominated by other pure strategies (e.g. “A” and “C” - depending on what action the other player takes, either one of these may be better).
The problem with pure strategies in a repeated game setting is that they're predictable, and can therefore be taken advantage of. In the example, if the row player always plays “A”, then the column player can adapt to always play “E”, and win. Of course, then the row player could adapt to always play “C”, which would result in the column player always playing “D”... Pure strategies aren't necessarily stable – you can't necessarily look at a game and say “always play A” (which is good, because that would be dull, and put me out of a job AND a hobby).
A mixed strategy for a given player is an assignment of probabilities over pure strategies – for example, a mixed strategy for the above table could be (50% play “A”, 50% play “C”). What would the best response to this strategy be? In this case, the answer is "anything" – no matter what the column player does, their expected winnings are still the same. However, any deviation from (50% play “D”, 50% play “E”) can be exploited by the row player, so they may as well play the fully randomised strategy. This pair of mixed strategies forms an equilibrium.
The king of economists, John Nash, proved that any such payoff matrix has at least one mixed-strategy equilibrium. Finding it is sometimes easy, as in the above example, but for more complex games it's often quite hard. This is a bad thing for mathematicians, but a good thing for game designers, as a mixed-strategy equilibrium solves the game when all players are rational – this is the optimal strategy, so there's no more fun to be had figuring out how to win.
There are two caveats to the above that are worth pointing out.
Firstly, people aren't always rational. This is either because they don't care enough to work out the optimal strategy, or they're incapable of it because the game is too difficult. If you can work out what strategy they're using instead, you might be able to figure out a counter-strategy that's better against them than the equilibrium response (e.g. Rock-Paper-Scissors, and your superstitious opponent always plays Rock).
Secondly, people aren't great at coming up with sequences of truly random numbers in order to choose actions. As a result, it's possible to guess what their next action might be based on what they've done previously (e.g. Rock-Paper-Scissors, and your opponent won with Rock last move. They're more likely to play Rock for this move).
These quirks to the reality of Nash equilibria can be leveraged in gameplay. Being able to out-predict your opponent can be pretty fun. Rock-Paper-Scissors is too easy an example, but what if we ratcheted up the complexity of the one-stage simultaneous move setting enough to be interesting?
Blottosphere
The central mechanic for such a game comes from a 1921 game theory paper by Emile Borel, as an example of a game where “the psychology of the players matters”. Variations of this were later studied further and referred to as “Colonel Blotto games”. While the basic formulation is superficially an exercise in hypothetical frivolity (like most good maths problems), it has applications in a wide variety of fields involving competitive resource allocation. Also, unwittingly, game design.
The basic Blotto game concerns two warring generals fighting over three battlefields. Each has the same number of indistinguishable faceless troops, divided into six battalions of equal strength. They must allocate these six battalions among the three battlefields – at least one battalion must be allocated to each battlefield. The battles then take place, and the battlefield is won by the side that supplied the most troops, because that's how wars work (if equal, the result is a draw). Each general is trying to win as many battlefields as possible, because that's how generals work.
This is a symmetric one-stage simultaneous move game with ten actions available to each player, corresponding to the following allocations: (2, 2, 2), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), (1, 1, 4), (1, 4, 1), (4, 1, 1). You could draw this out as a 10x10 payoff matrix, but I won't because I'm lazy, and I'm not going to refer to specifics. It's just good to know that you can.
With some inspection, it's easy enough to see that the (1, 1, 4), (1, 4, 1) and (4, 1, 1) allocations are pretty bad – they always lose to (2, 2, 2), and they lose to any permutation of (1, 2, 3) where the 2 and the 3 go up against the two 1s. At best, they draw with any other (1, 1, 4) and some of the (1, 2, 3) permutations. As such, they're dominated by the other strategies, as the other strategies can't do worse than them, but can do better. If we throw them away, you're left with seven possible actions.
If both players use the pure strategy (2, 2, 2), the result will be a draw. In addition, the result will also be a draw if either player switches to any of the (1, 2, 3) permutations – they have no incentive to do so. This makes [(2, 2, 2); (2, 2, 2)] a (weak) pure strategy equilibrium. This is actually one of an infinite number of mixed strategy equilibria corresponding to either player using a strategy of the (2, 2, 2) action with probability p, and each of the (1, 2, 3) actions with probability (1-p)/6.
Even though this case has a boring pure strategy solution, the mechanism of simultaneously allocating resources to locations has some merit – especially as you increase the number of units that can be allocated. Even if we restrict the action space to just permutations of (1, 2, 3), the game morphs from having a simple pure strategy into a slightly warped Rock-Paper-Scissors – each permutation has four opposing permutations that draw with it, one that beats it, and one that loses to it. This still isn't great, but it's a start. How can we make this more interesting?
The main problem the Blotto game has is symmetry. Each battlefield is worth the same amount, and each player values each battlefield the same. The second problem the Blotto game has is that all this symmetric information is public. Each player knows how the other values each battlefield. By changing both of these things we're essentially adjusting the payoff matrix – instead of tidy symmetry, we're deliberately making it asymmetrical and harder to calculate, which in turn will make it harder to calculate the optimal strategies (Nash equilibria). My hope is that this will be beyond the bounds of what normal players will attempt through regular play, and thus the opponent modelling skills alluded to previously will become more important when opponents adopt suboptimal, predictable strategies.
Enough maths. On with the game!
Thieves' Blotto
This is played with a deck of 18 cards. You can make your own (and with further testing I may make a printable version), but it might be easier for now to take them from a deck of standard playing cards (hey, remember those?). You Will Need:
J, Q, K of two suits (e.g. spades, hearts)
4,5,6 of spades
3,4,5 of hearts
2,3,4 of diamonds
2,3,4 of clubs
The J, Q, K of one suit are given to each player. These are your bidding/allocation cards, corresponding to the Blotto allocations 1, 2, and 3. (Or, to put it another way, K beats Q and J; Q beats J; J beats nothing)
The remaining cards are shuffled and placed in a pile. These are the objects you'll be bidding on.
The game consists of four rounds. In each round, take the top three cards of the pile, place two of them face up in the centre of the table, and place the third face down in the centre of the table.
Each player then places one of their bidding cards next to each object, face down. Once both players have done this for each object, reveal all six bidding cards. For each object, the player who bid the higher amount wins the object – if the bids are the same, then that object is discarded face down.
After four of these rounds, each player will have amassed up to 8 cards. Each player's score is calculated as follows:
Each object is worth its face value (e.g. 5 of hearts is worth 5 points)
Having two cards of the same suit is worth an extra 1 point (e.g. 2 and 3 of clubs are together worth 6 points)
Having three cards of the same suit is worth an extra 3 points (e.g. 4,5,6 of spades are together worth 18 points)
The player with the highest score wins, as usual.
Design Explanation
The asymmetry of the standard Blotto setup is ensured by making each object worth a different number of points. This is compounded by the suit bonuses, which after the first round make each object worth different amounts to each player. The hidden card each round obscures the score of your opponent and makes it harder to assess how they value each subsequent object (because of suit bonuses) – technically, any hidden card is a uniform superposition of all hidden cards at any point in the game (ignoring possible inferences from previous rounds – I'm not sure how significant this will be), so you can use this blended value in each round's payoff matrix, but this is a difficult operation for people to perform on the fly, meaning they should fall back to exploitable heuristics.
Adding in this asymmetry/hidden information does mean that the (2, 2, 2) and (1, 1, 4) (etc.) allocations now serve valid purposes – however, it's hard to represent these elegantly with cards like you can do with (1, 2, 3), so I dropped them. It's not a major loss.
Why call this "Thieves' Blotto"? I wanted to add flavour, somehow, and didn't like the combative battlefield stuff present in the initial formulation of the game – it's overdone, and didn't quite fit the mechanics of permanently acquiring objects without losing your bidding cards. Having the bidding cards be thieves of different ranks and the objects being valuable things to steal seemed more appropriate, and also helped with the flavour of the face-down object. If this tests successfully enough to warrant a printable version, this is probably the concept I'd use. (also considered: pirates, cowboys, monster hunters)
Conclusion
I like the simultaneous bid mechanic at work here – as long as the objects being bid on have some complexity to their value, this rewards reading how your opponent values things and predicting what they'll do. It feels like this could easily be a component of a larger game, and it's something I'll squirrel away for the future.
Still, as I promised in the intro paragraphs, I need to branch out from this kind of stuff in order to recruit more playtesters. To this end, I have most of another post written on player taxonomies – can we put people into boxes based on their game-playing motivations, and if so, what are they? It's an interesting area that's coincidentally math-free – I should have it up in the next few days.