Ridere, ludere, hoc est vivere.

Wednesday, February 22, 2017

Notes on Games of Strategy

Over three years ago, I wrote about my effort to approach a simple three-player race game using game theory.  Economist and game designer Dr. Aaron Honsowetz responded, which led to his recommendation that I look up the book Games of Strategy by Avinash Dixit, Susan Skeath, and David Reiley.  I finally obtained the third edition recently, and that has led Aaron, fellow designer Austin Smokowicz, and I to explore Dixit Skeath and Reiley's text in a kind of virtual book club.

Thinking about strategic games

We live-streamed last night's discussion.  We start with the first two chapters, which motivate the study of game theory and then define some terms and categories. 

Strategic games depend on player decision-making, as distinguished from games of chance, which depend on luck, and from games of skill, which depend on proficiency, dexterity, quickness of mind, or practice.  So, for example, chess is a game of strategy whose outcome is determined by the decisions of the players.  Bingo is a game of chance, whose outcome is determined by the order of randomly selected numbers.  Bowling is a game of skill, whose outcome is determined by the strength, accuracy, and proficiency of the bowlers.

The authors distinguish decisions that people make that are independent of the decisions of others from strategic games, in which people know that the results of their decisions depend on the decisions of others as well, i.e. that their decisions are interactive.  So, for example, blackjack plays out based on the decisions of each player in isolation, regardless of the other players at the table and of the dealer, who follows strict rules and makes no independent choices.  Poker, by contrast, plays out based on the interaction of the decisions among players.  So poker meets the definition of a strategic game, while blackjack does not.

The authors proceed to classify games in anticipation of the structure of the rest of the book:
  • In sequential games, players make decisions whose immediate outcome is unaffected by other players, as opposed to games with simultaneous moves, in which players must anticipate the unknown decisions of other players.  Chess is sequential, while rock-paper-scissors is simultaneous.
  • Some games pose players with strictly conflicting interests while others might involve common interests among the participants.  So most wargames are strictly conflicting, while Dead of Winter introduces both common and individual goals.
  • One-time games are distinguished from repeated games with the same opponents, which in turn are distinguished from games involving changing opponents.  So a one-time-only game might be the courtship, engagement, and marriage of a couple.  A bridge club plays the same game with the same opponents repeatedly.  And a single-elimination game tournament generally involves playing the same game with different opponents in each session.
  • The authors define games with full and equal information vs partial or unequal information:  External uncertainty applies to unknown information independent of others' decisions.  Strategic uncertainty applies to the unknown decisions by others.  A game with either or both has imperfect information; a game with no such uncertainty has perfect information.  A game in which one player has more information than another has asymmetric information.  So for example, a card game with a shuffled deck involves external uncertainty.  Rock-paper-scissors involves strategic uncertainty.  Chess is a perfect information game.  Scotland Yard has asymmetric information.  
Strategies that share or reveal information deliberately are called signaling.  Strategies that seek to motivate an opponent to reveal information are called screening. We spent quite a bit of time discussing examples of signaling, such as playing chicken and throwing the steering wheel out the window to demonstrate to an opponent your commitment not to swerve.  Aaron cited an example from Euchre in which card play signals to a partner information about your hand or about what the opponents do or don't have.  To Aaron's point, the important component of signaling is the degree of commitment to a decision.  Austin used an example from Hanabi as a method of communicating intent to fellow players.
 After the chat, Aaron further refined our discussion of signaling: 
Bluffing itself is an uncreditable signal. You claim you are committing to something (to alter player behavior).  To be a creditable signal, the price to make the signal must be sufficiently high that only a person committed to the action is willing to pay it. When I remove my steering wheel and toss it out the window I am saying I am willing to drive straight no matter the price. If the price of making the signal is too low, than people can make the signal without creditably committing to the action (bluffing) and destroy the ability for the signal to indicate you are going to take a particular action.
Screening involves eliciting information, and one way of doing that is to make an offer like a trade in Catan to see how players respond and thereby gain information about the contents of their hand and to an extent their future intentions.
During our discussion, I provided an example of sequential interactive decision-making that led to another follow-up by Aaron regarding signaling and screening.  In a game of Agricola, I built fences in anticipation of an opportunity to take sheep.  An opponent, who had no need of sheep and no means to store or cook them, took them anyway and let them run free to deprive me of that opportunity.  Said Aaron,
Your sheep example from Agricola was (if there was no other value for that many fences) a credible commitment that you would take sheep if they were available.  If that is the case it could also be part of a screening tactic.  If everyone is paying attention, it forces the last person before you who has a lower value play (assuming you are competitive in the game) to reveal that by responding to you.  And while you may not get the points from the sheep, it may still have been the best move because it blocked an opponent from taking an action to advance their score.
  • Games can have fixed vs manipulable rules.  Most tabletop games with which we are familiar have fixed rule-sets.  I imagine games with manipulable rules to include the interactions within a legislative body, whose rules of order may be modified by a party in power. 
  • The book defines cooperative games differently from the conventional use of the term that modern game-players might know.  The authors refer to games in which agreements are enforceable as Cooperative, while games with unenforceable agreements and that allow players to act in their own best interests are called non-cooperative games.  By these definitions, Catan is cooperative, inasmuch as a trade is an enforceable agreement; both parties are required by the game rules to hold up their end of the bargain.  Diplomacy is a non-cooperative game, since a player can commit to a future action and then renege on that commitment. 
Terminology and other concepts:
  • Strategies are available choices, or more generally a set of guidelines or algorithms by which individual decisions are made - a plan for a succession of actions in response to evolving circumstances, presumably due to the actions of other players.
  • Payoffs are the outcomes of interactive decisions, including expected payoff based on a probability distribution of random outcomes.
  • Rationality, or rational behavior, assumes perfect calculating players that consistently follow the best strategy pursuant to a completely known self interest.
  • Games involve a common knowledge of rules, specifically knowing who the players are, their available strategies or choices, the payoffs for each interaction of strategies, and an assumption of rational behavior.
  • When rational players interact, the game reaches an equilibrium where by each player is using the strategy that best responds to the other players' strategies. 
  • As opposed to assumed perfect rationality and calculated equilibrium, an evolutionary approach to games allows for a dynamic process in which poor calculators are motivated to choose strategies that proved more successful in previous plays of the game through observation, imitation, and learning.
  • Observations and experiments can help structure game theory and provide a check against its results.
The book stipulates that game theory can help to explain observed behavior of interacting decision makers, predict likely choices of rational actors, and prescribe strategic decisions.

I had one question that did not get addressed in our chat:  Can a game have more than one equilibrium, i.e. can there be local points of optimization that could emerge in an evolutionary approach different from what an analysis of perfect strategies would indicate?  The answer may come up in later discussions.

Next we will explore Chapter 3, "Games with Sequential Moves."

Sunday, January 15, 2017

Papal Pilgrimage: A preview of the sequel to Avignon

The microgame format that Love Letter popularized poses a considerable design challenge.  Fewer cards mean players face statistically fewer different situations.  Pared down to a skeletal structure, a microgame really has to make every card significant and capitalize on every opportunity for interaction. John du Bois introduced a clever two-player tug-of-war in this format with the 2016 Button Shy game Avignon: A Clash of Popes.  To that tight little design Button Shy Games is Kickstarting a sequel, Avignon: Pilgrimage, that introduces new characters that can stand alone as a separate game or that players can mix in with the original Avignon for a variety of interactions.

The games are set during the Western Schism of the Catholic Church in the late 14th century, when rival popes in Avignon and Rome sought to wrest control of the Church from one another.  Each player represents one of the papal factions recruiting key figures into the supporting congregation of Avignon or Rome to solidify that pope's control.  

Avignon in both its original format and its sequel constitutes a logic puzzle of card interactions with the goal of moving character cards to one's own side of the table.  Available actions are simple.  The complexity comes in the individual character card "Petition" actions that manipulate the positions of cards and the victory conditions.  The new sequel, Avignon: Pilgrimage, introduces a number of mechanics relying on the relative positions of other characters on the table and thereby adds a new layer of interactive depth.

The five locations on the road to papal legitimacy
between Avignon and Rome.  The Ascetic is shown in
Genoa, which is where he really wants to be.
The game consists of twelve cards, each depicting one of six characters.  Five cards are dealt in a row on the table between the players.  The starting row represents Genoa.  A card can be moved from Genoa in one direction to Florence, to Rome, and then to the Roman player's congregation.  Or it can move from Genoa in the other direction to Nice, to Avignon, and then to the Avignon player's congregation.  The goal is to collect three characters into one's own congregation - or to meet the special victory condition of a character in play.

Available actions to a player are to pull a character one space closer, to push a character one space further, to excommunicate a character (removing it from its location and replacing it with another character from the draw deck), or to petition a character (activating its special ability).  The crucial rule in Avignon is that although a player takes two actions per turn, they must be different actions.  So only one character can be petitioned for its special effect; only one character can be pulled; only one character can be excommunicated.  So good game play consists of recognizing and implementing the right combinations of actions to gain ground on the opponent in a lasting way.

Some Pilgrimage characters seem more interesting than others.  Petitioning the Courtesan pulls a character whose position is closer to the opponent than is the Courtesan; so her position generally benefits the player to whom she is closer.  The Canonist allows switching the positions of two characters, then pulling one and pushing the other.  Petitioning the Nuncio pushes him one space and then pulls all other characters that are in his new location - essentially having him trade places with everyone else in that location.  

Other characters seem less interesting.  Petitioning the Vicar pushes a Vicar (or Bishop from the original Avignon) two spaces, excommunicates another character, and pulls that character one space - a move that seems to make a considerable position sacrifice in exchange for a random character replacement.  An Ascetic ending the turn in Rome or Avignon automatically moves one space closer to Genoa, making him particularly difficult to recruit (and thereby less interesting).  The Scribe allows a player whose turn starts with the Scribe in Genoa to win with only two characters in his or her congregation.  This effect can be a game-ending (and random-feeling) consequence of pulling a character into one's congregation only to see that character replaced by the Scribe in Genoa to start the opponent's turn.

Having played Avignon: Clash of Popes and Avignon: Pilgrimage separately, I look forward to mixing them together and seeing the cards from both sets interact with each other.  I expect the logical conundrums to deepen even further, and I always enjoy when a small-footprint game turns into a little brain-cell-burner.

Unlike many microgames, there is no hidden information (beyond the draw deck) and no bluffing, so this is not a social game of getting inside the opponent's head.  People who like quick microgames with real logic challenges will like both Avignon games.  Pilgrimage by itself is a bit more dynamic but also a bit uneven relative to the original.  All in all there is a lot of game in this small footprint.