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 me 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 whereby 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."