Ridere, ludere, hoc est vivere.

Thursday, November 22, 2018


Since the signing of "Magnificent Marvels" in August, I had resumed posting weekly here about our design progress, about first impressions of various new and upcoming games, and especially about notes on game theory. At this point, more pressing opportunities demand my attention, so rather than continue to assemble an essay every Monday, I intend to shift my focus to other avenues of board game hobby exploration and creativity. I am grateful this Thanksgiving for my readers and their feedback. I will remain active on Twitter at @PaulOwenGames. Until another time when I have more to share, I wish everybody the best.

Monday, November 19, 2018

Notes on combining sequential and simultaneous moves

As part of a series of discussions on Games of Strategy, I've written here about games with sequential moves - those in which players "take turns" and each decision is made with full knowledge of the opponent's last decision - as well as games with simultaneous moves - those in which players make decisions not knowing which option an opponent has selected. Continuing our exploration of game theory, Dr. Wictz and I further discussed games that combine sequential and simultaneous moves.

Game tree for "Battle of the Sexes"
Source: "Managerial Economics Online,"
A simultaneous move pure strategy game can be converted into a sequential move game, which can change the outcome depending on the payoff matrix. When "The Battle of the Sexes" is a simultaneous coordination game (like "Will Harry Meet Sally?"), there are two equilibria, and the outcome is indeterminate. When it is sequential, the player moving first will select the option they prefer, and the player moving second will follow suit because it becomes the only beneficial option. The result is a first-player advantage, meaning that the player moving first maximizes their payoff.

Suppose the tennis match we looked at in our examination of mixed-strategy games were converted into a sequential move game, perhaps because the defender can react quickly enough to defend either shot regardless of what the shooter decides to do. Then, by rollback analysis, the shooter will be motivated always to choose the shot that the defender is less effective at defending, and then the defender will defend it. The result is a second-player advantage; the first player can never maximize their payoff because the second player can always respond. If, on the other hand, a slower defender has a "tell" and commits to a defense before the shooter decides on their shot, then the shooter can always attack the weak side of the defense and maximize their payoff - again, a second-move advantage.

Recalling that we can model sequential games as a decision-tree network, we then can represent each node of the sequential game as a simultaneous game decision matrix whose outcome determines the next branch of the sequential game. We discussed The Resistance, a hidden-identity game that consists of a sequence of simultaneous-decision games. The decision tree extends over five rounds in which each team seeks to win three rounds. Each round can itself be thought of as a sequential decision tree of simultaneous decisions - first voting whether to accept a mission team, and second among those on the team, deciding whether to support or sabotage the mission.

A sequential game can have an equilibrium path that can be determined by roll-back analysis - pruning non-optimal branches of the tree from the end backward to the beginning. I described this processes in my discussion of bidding and game theory. But a sequential game can also proceed along an "off-equilibrium path" if players make non-optimal decisions - more common in complex games where the decision tree or simultaneous decision matrix defies complete analysis.

A subgame is defined as part of a multi-move game that begins at a particular decision-tree node of the original game. A "subgame-perfect equilibrium (SPE)" is a set of strategies - one for each player - that defines the optimal decision at each node in the decision tree, whether or not that node is on the equilibrium decision tree path. No matter where the game may start, for every subgame, the SPE defines the optimal strategies for both players.

Considering our Resistance example, information is incomplete, so players generally will signal intent in attempts to steer one another along a decision path to a subgame that is closer to their own best result. The success of those attempts depends on the credibility of the signal, and therein lies the heart of such social interactive games.

Monday, November 12, 2018

Top ten games with my wife - 2018 edition

Four years ago I was inspired to identify the top ten games that my wife and I liked to play together. We've played a lot of games since then, and added a lot of games to our collection. So I thought I'd revisit this question. We found it very difficult to narrow the list down to ten; there are a dozen not listed here that we could easily have included instead.

Monday, November 5, 2018

Highest-rated out-of-print games - 2018 edition

Four years ago I identified the top five out-of-print games on boardgamegeek. Since then, a number of games have found new life as reprints, so I thought the topic worth revisiting to see what's "hot but hard to find."

Monday, October 29, 2018

Notes on simultaneous-move games with continuous strategies

My recent posts on game theory have focused on games with discrete strategies, which is to say that players are faced with a finite number of choices for each decision. Some time ago, in our colloquium on games of strategy, Aaron Honsowetz, Austin Smokowicz, and I discussed games with continuous strategies (also on video) - those games in which players choose a value on a spectrum, such as a price to set on a commodity. 

Monday, October 22, 2018

European board games before Catan

Game designer Rob Newton asked on Twitter what people would consider "classic literature of the board game world." Fellow designer Jonathan Weaver responded in three categories - citing Chess and Go as "ancient literature," Monopoly as "classic American literature," and then added, "whatever the predecessor of Catan is would be classic European literature." Some time ago, I was inspired to go back and look at what board games Americans had available to them when Monopoly arrived.  Now I was faced with a similar question for Europe before Catan. I already knew that Clue was originally English and Risk came from a French design, but Weaver's response made me realize that I really couldn't identify other European games released prior to 1995. Surely Catan couldn't have been the first popular European board game, so I felt that some self-education was in order.

Monday, October 15, 2018

Notes on games with mixed strategies

In an earlier post regarding theory of simultaneous move games, I concluded with an example of a game between two tennis players that did not demonstrate a Nash equilibrium between its two pure strategies. Sam Hillier: Consulting Philosopher more recently elaborated on the topic with an excellent post on mixed strategies. Whereas I had approached the question of an equilibrium for a single tennis shot and concluded that none existed, a tennis match of course includes many shots, so players have an opportunity to invoke a weighted mix of shots and defenses between the two options.

Monday, October 8, 2018

Spies vs spies: First impression of Cold Warrior

I have long felt that the Cold War represents a rich thematic opportunity for board games. Yet I can identify only a handful of games set in that NATO-Soviet contest of diplomacy, espionage, and brinkmanship that rank in the top 1000 of boardgamegeek.* Into this cloak-and-dagger arena the new designer Wes Crawford introduces Cold Warrior (artist Jimmy Malone, published on Game Crafter).