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Games and Actions

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Part of the Synthese Library book series (SYLI,volume 354)


Our main focus in this book has been the representation of preferences of one agent, their entanglement with informational attitudes like beliefs, and the various events that dynamically change these informational and evaluative attitudes. But like all aspects of agency, such single steps are the building blocks for something larger, viz. interaction between different agents over time. For instance, a learning process is typically a long-term history where an agent interacts with a source of information, and where both knowledge, beliefs, and goals may change over time. At its most pregnant, this longer-term interactive aspects occurs in games, where many of our main concerns occur very concretely – including significant meetings between information and evaluation.


  • Game Model
  • Strategy Profile
  • Game Tree
  • Strategic Game
  • Priority Graph

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Fig. 12.9


  1. 1.

    For more information about Backward Induction and its history, see [147].

  2. 2.

    I thank the authors for their permission to use relevant material from their survey.

  3. 3.

    More abstract worlds might carry strategy profiles without being identical to them.

  4. 4.

    cf. [144] and [45] for formal details.

  5. 5.

    There is a much more literature on these topics. cf. [25, 26, 106, 56], and [137].

  6. 6.

    Rationality is a sweeping form of “entanglement”, a bridge law between what we know or believe about the outcomes of our actions and how we evaluate these.

  7. 7.

    Here \(move_i=\bigcup_{a\ is an i-move}\) a, turn i is a propositional variable saying that it is i’s turn to move, and end is a propositional variable true at only end nodes.

  8. 8.

    The definition of S can be stated in a well-known logic of computation, viz. first-order fixed-point logic LFP(FO) [70]. The dissertation [82] has details on this way of defining game solution methods.

  9. 9.

    The following section mainly follows the topics and results of [38, 82].

  10. 10.

    There are also “self-refuting” propositions, becoming false everywhere in the limit model of their repeated announcement. This happens, for instance, with the repeated ignorance assertions in the Muddy Children puzzle (cf. [73]).

  11. 11.

    Game-theoretic solution procedures have been applied to moral deliberation in [136].

  12. 12.

    Deep comparisons are reminiscent of how one computes a probabilistic expected value.

  13. 13.

    This has a technical proviso: These leaf orders must be “node-compatible”: cf. [82].

  14. 14.

    Concretely, one might do this as follows, taking a cue from the DEL event models in Chapter 2. Start with models for PDL with preference between states. This interprets a standard dynamic language plus a preference modality. Now let actions by themselves – or better, single transitions between states as in Arrow Logic (cf. [23]) – form a modal model, with a binary betterness relation ≤ between transitions that can have unary atomic properties \(p, q, ...\) The matching double modal language now also has formulas expressing properties of actions. One useful modal operator \(END\varphi\) would say about a transition (arrow) a that the state-formula φ holds at the end-point of the a. And a second preference modality \(\langle<\rangle\psi\), now on actions, will say that transition a sees a better transition b satisfying ψ. The logic will now also have axioms relating state and action modalities, depending on how one sees between preferring actions and preferring their outcomes. Backward Induction was one way of creating such links, but there may be others.

  15. 15.

    Their slogan is that we need a “Theory of Play” going beyond game theory.

  16. 16.

    Quantitative forms of diversity occurred in Chapter 6 with weighing rules for past experience and current observation. It is worthy of mentioning that [3] and [4] studied behaviors of resource-bounded agents.

  17. 17.

    Interestingly, from a deontic point of view, we made both players better off by restricting the freedom of one. Conversely then, an increase in freedom is not always a good thing.

  18. 18.

    An interesting further issue is how public announcement changes effects of complex strategies in a game [30]. Consider the strategy modality \(\{\sigma\}\psi\) saying that playing σ always leads to end points satisfying ψ. Here is a valid reduction axiom for reasoning about effects of strategies in the changed game: \(\langle!\varphi\rangle\{\sigma\}\psi \leftrightarrow (\varphi \wedge \{\sigma \mid \varphi\} \langle!\varphi\rangle\psi\). Its notation \(\{\sigma \mid \varphi\}\) refers to an obvious “relativization” of a PDL program σ to the submodel defined by φ.

  19. 19.

    Impossibility can also be epistemic. I thought that I could reach the fruits and steal them. Now I cannot, and I think they have probably been sour anyway. This phenomenon has been discussed in decision theory and philosophy, cf. [72] on “sour grapes”.

  20. 20.

    Indeed, given that Backward Induction is played on a certain set of preferences, one can rationalize the given preferences, in the sense of our earlier discussion, to new very special preferences that simplify the reasoning. For instance, in our initial running example, one might make the actual outcome the very best for all without changing the BI strategy.

  21. 21.

    We leave the simple proof of correctness to the reader here.

  22. 22.

    For instance, it fits in the PDL-program format of Chapter 4.

  23. 23.

    The axiom for the move modality \(\langle a\rangle\) is a simple operator commutation. A full version would need a small extension to “relativized upgrades” mentioned above.

  24. 24.

    This time, the given σ A never prescribes a move that is strictly dominated assuming that further play proceeds via the given strategy for A and any moves for E.

  25. 25.

    Note the analogy with the Confluence Property pictured in Section 12.4.

  26. 26.

    Reference [31] explains A’s “irrational move” of going right in our running example in terms of running risks for the common good, expecting E to return the favor.

  27. 27.

    A concrete DEL-related use of goals is made in the “knowledge games” of [1].

  28. 28.

    Alignment can be very hard to judge if we just have players’ extensional comparison lists of outcomes: just as we cannot say much about voters’ similarities or dissimilarities in thinking if we only look at their voting records.

  29. 29.

    Another relevant case would be the infinite games of evolutionary game theory : cf. [108].


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Correspondence to Fenrong Liu .

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Liu, F. (2011). Games and Actions. In: Reasoning about Preference Dynamics. Synthese Library, vol 354. Springer, Dordrecht.

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