A good understanding of Nash equilibria is necessary for understanding the implicit auction theory behind most games. Presh Talwalkar has written a rather nice casual description of them in his Mind Your Decisions blog in the artcle, The Non-Mathematical Guide to Fixed Point Theorems and Proving Nash Equilibria Exist. He similarly casually and usefully discusses Arrow’s Impossibility Theorem (central to thoroughly understanding any auction or voting system) and Pareto Efficiency in another article, Arrow’s Impossibility Theorem and The Voting Paradox. Good stuff.

A few other interesting articles from a game-design perspective:

• Strategic Commitments: How to Lose Weight and Live up to New Years Resolutions
• Why Patience Pays Off in Negotiations
• Voting Power in Israeli Judge Selection and the Shapley-Shubik Index

In all the recent discussions of Temporary Emergent Alliances (or Manipulating Others’ Incentive Structures), Wabash Cannonball, Pampas Railroads, Muck & Brass and similar games the following article fell across my browser: Game Theory in The Dark Knight: A Critical Review of the Opening Scene (Spoilers). Several of the other articles in the author’s Game Theory Series are also interesting.

Wabash Cannonball Expansion - Erie Railroad belies its effect on the game with its small size. The Erie RR is nitroglycerin in a cocktail shaker. It makes the already rather delicately knife-edged game of Wabash Cannonball even more tetchy with the penalties for mis-steps ever larger and the path ever narrower and risk-fraught.

The Erie expansion consists of a single new Erie company with a single share, 14 cubes (1 income marker, and 13 track) and a slip of paper containing the few special rules governing when and how the Erie becomes available for play. More specifically it has enough cubes to reach both Detroit and Chicago.

The Erie RR becomes available for Capitalisation when any of four specially nominated cities have track built in them by another company. The Erie’s home station is roughly in the middle of the northern edge of the board. It has no special rules except that:

• It can build into New York City at a cost of $6 for an income boost of $8
• The N-companies-out-of-shares game-end condition is increased from 3 companies to 4 companies

Functionally the Erie RR is simple enough except in the implications:

• The Erie can only be Capitalised by explicit choice. Why a player would Capitalise the Erie is a difficult question:
  • Because the acting player can win it and profit
  • To tempt another player into a posture that can then be wielded against them (push a short game against a player positioned for a long game and visa versa)
  • To drive an early/faster game-end
  • To tempt another player into sundering their extant alliances
• The player that wins the Erie:
  • Is almost certainly no longer a functioning member of any prior alliances as their incentives now orbit the Erie RR due to their high cash investment in the Erie RR
  • (Likely) explicitly postured for a long game but with greatly reduced ability to influence game length
• The Erie RR is more valuable to owners of NYC RR and Pennsylvania RR shares
  • The Erie is most likely to share cities with the NYC RR and second most likely to share cities with the Pennsylvania RR.
• Once the Erie RR opens the Development actions becomes much more interesting to all players
  • Once the Erie RR player runs out of Expansion actions, or runs out of cash, very likely their most attractive action is Development
  • If the Erie RR player can benefit multiple of their companies with Development (thus the interest in the NYC RR and the Pennsylvania RR), then so much the better.
  • Due to the steady depletion of Development actions by the Erie RR player, the Development action becomes more attractive for the other players. Their shares are less diluted with the removal of the Erie RR player’s maniacal focus on the Erie RR and Development is thus more likely to lead to a direct/tangible cash income increase for them.
  • Increased use of Development as versus the standard routine of Capitalise/Expand affords control of additional control game length and control of who is the start player for the next round.
    • It becomes easier for low player count games to end due to track cube exhaustion
    • It becomes easier (albeit likely still rare) for middle-player count games (ie 4 players) to end due to Development cube exhaustion.
    • Exact control of game length is more diffuse while the Erie RR player’s maniacal focus encourages all the players to posture more heavily toward a specific game length
• Detecting when it is no longer viable to invest in the Erie RR versus when it is too soon (not enough cash or game-control) is delicate and fraught with failure
  • The Erie RR will likely need somewhere between $20 and $30 (possibly more) in order to fully afford its own expansion opportunities
  • Due to the single share all capitalisation must occur from the initial share sale. Forest development is an unviable capitalisation source.
  • Whether or not the Erie RR will get those opportunities is uncertain
  • As only one player can Expand the Erie RR, company income growth is necessarily slow as compared to collusive Expansion alliances among other players
  • However due to lack of dilution, income growth is fast(er)
    • With a large caveat for Chicago Dividends
    • Those same Chicago-bound alliances encourage the Erie RR player to sabotage the other RRs by running them short of cubes
      • The opportunity cost of not spending that Expansion on the Erie RR is high

The Erie RR is highly attractive due to the massive potential profits. Should it get to both Detroit and Chicago it can easily pay $25/share! However as noted above this great profit potential comes with risks for all the players, not just the Erie RR-investing player. That balance of risk and reward forces the players to begin positioning themselves in relation to the Erie RR from the initial auction of a Pennsylvania RR share at the start of the game and every portion of the game after that becomes a question that also needs to be evaluated in relation to the Erie, whether or not it is in the game yet.

I recommend Wabash Cannonball Expansion - Erie Railroad only to experienced Wabash Cannonball players. Unless the players comprehend not only the base game’s arithmetic and alliance system, but also how to posture against game length and how to adroitly wield game-length control against the other players then the Erie will simply walk all over them and render the game results an opaque who-goofed crapshoot. With skillful players however it becomes a delightful game of balancing on razorblades and juggling waterballoons of nitroglycerin. It is quite the designer tour-de-force. Amazingly subtle and pervasive. It is rare that so little turns so much of a game on its ear without also breaking it. Everything is the same and yet different. Bravo!

I primarily use the Remainder Game for start player selection but there’s no inherent reason it has to be limited to that application. The pattern is simple:

1. One player calls for the start player selection, Everybody stick out some fingers. 1, 2, 3!
2. Take the modulus of the total number of fingers shown by the players by the number of player and starting at zero, count out that many players in rotation from the calling player.
3. The indicated player is the start player.

The Remainder Game is efficient, deterministic results (no repeats, roll-offs or ties possible), actually random¹, short constant execution time, works with any number of players in any situation, works with any game in any situation, clearly auditable by all concerned, and requires no equipment.

Ariel has come up with an impressive follow-on T-shirt suggestion:

Chris Johnson (fnord23) kindly noticed an error in my charts for Pampas Railroads. Corrected versions are in the normal place.

I previously simplified the implemented definition of a future down to:

1. Buy goods now to be delivered at time X
2. Sell goods now to be delivered at time Q

That was a mistake. A necessary element of the core problem of Modern Mogul is that fact that the players should (nearly) never be playing the game as it exists in present time. Instead they should be playing the game as they hope it will be some number of turns in the future. In the above simplified definition the capital is both committed and spent now and only satisfaction is delayed. The more interesting and useful form has the commitment occur now and the capital expenditure when the future matures. In this way the players will not only have to juggle the vagaries of the future market, but also the vagaries of their future liquidity as a result of market activity between then and now. Much better.

Attempting a taxonomy of negotiation:

Open Negotiation

Free-form done through conversation and persuasion. Open Negotiation is unbounded/unconstrained and is frequently only incidentally related to direct/current in-game concerns (what tokens are where, what permissions are granted etc) and is generally much more interested in establishing consensus on mutual policies and long-term collusive patterns. Diplomacy is perhaps the titular grandfather of Open Negotiation in game design. The key elements are primary focus of the negotiation is to persuade another player to pro-actively collude (Will you do XYZ for me?), and while the focus of the negotiation is informed by the game, the majority of the negotiation content is meta to the game details.

Simple Negotiation

Frequently tactical and usually directly informed by and concerning current game specifics and the disposition of specific game tokens. The most common forms are simple requests for permission or abeyance. May I do QRS? or If I do ABC will you not attack me? The key element is direct focus on and discussion of game particulars (and thus a rather tactical focus). Questions of policy and long-term collusive patterns are not part of Simple Negotiation.

Implicit Negotiation

In short, where moves are offers. Moves, independently arrived at by the players, without discussion, for negotiative patterns due to common understandings of collusive self-interest which emerge from the game-play. Gun-boat¹ no-press² Diplomacy is the great-grand-daddy here. More recently Wabash Cannonball has epitomised this model[^3].

Personal views

I am generally not a fan of Open Negotiation in games, rather like Simple Negotiation as (exemplified by such games as Lords of the Spanish Main, Quo Vadis and Traders of Genoa) and adore Implicit Negotiation (eg Wabash Cannonball, Pampas Railroads, or King of Siam).

A while back Jacob Butcher finally categorised me, You like negotiation games! He’s right of course and I especially like them when the negotiation is implicit instead of explicit, or to quote a more recent conversation with another friend, Where your moves are offers.

I love that idea. Moves are offers. Adorable. Juicy. It is likely that single characteristic, married tightly to a raw game theory core¹, is what has so attracted me to Wabash Cannonball. Moves as offers.

So I’ve been thinking for about a week now, mulling around ideas surrounding moves as offers and trying to see what sort of game could be wrapped on that skein. The working concept, which is weak, is a game in which each player has 1/Nth of a limited resource in the game and every move they either offer (partial) access to their stock to one or more other players, or exploits the access that others have offered while also offering access in return. All terribly vague, but the core sense is a sort of dance of offers and engagement and commitment and perhaps even betrayal. The obvious comparables are So Long Sucker and Intrige, but I’d like something far more implicit than those fine stalwarts, as well as far more implicit than explicit.

So far I’ve come up with…nothing. Which is fine for this stage as the thought toy has been delightful, but I’d would like to tangible realise this idea. Sadly inspiration has fled. Still, the thought toy is so charming, so very Mary Poppins in every possible very nearly perfect way.

I’m not getting to play with Muck & Brass as much as I’d like lately – Lords of the Spanish Main is occupying too much of my attention – so a little disclosure seems in order: Rules for Muck & Brass!

I also give you the brief graphical evolution of Muck & Brass.